Ramsey theory and small countable ordinals Andr es Eduardo Caicedo - - PowerPoint PPT Presentation

Ramsey theory and small countable ordinals

Andr´ es Eduardo Caicedo

Mathematical Reviews

Mathematics and Computer Science Colloquium, Albion College, April 13, 2017

Caicedo Ramsey theory and small countable ordinals

Introduction

My interest in Ramsey theory started as an undergraduate, reading a set of notes by Ronald Graham (a 2nd edn., coauthored with Steve Butler, was published in 2015), and the book by Graham, Bruce Rothschild, and Joel Spencer (2nd edn., 1990).

Caicedo Ramsey theory and small countable ordinals

My favorite anecdote to motivate Ramsey’s theorem is a story at the beginning of Extremal and Probabilistic Combinatorics, by Noga Alon and Michael Krivelevich (The Princeton companion to mathematics, Timothy Gowers, June Barrow-Green, Imre Leader, eds., PUP, 2008. Chapter IV.19, pp. 562–575).

Caicedo Ramsey theory and small countable ordinals

In the course of an examination of friendship between children some fifty years ago, the Hungarian sociologist Sandor Szalai observed that among any group of about twenty children he checked, he could always find four children any two of whom were friends, or else four children no two of whom were friends. Despite the temptation to try to draw sociological conclusions, Szalai realized that this might well be a mathematical phenomenon rather than a sociological

• ne. Indeed, a brief discussion with the mathematicians

Erd˝

• s, Tur´

an, and S´

• s convinced him this was the case.

What Szalai is observing, in the language of Ramsey theory, is that r(4, 4) ≤ 20, that is, any graph on 20 vertices either contains a copy of K4, the complete graph on 4 vertices, or a copy of ¯ K4, the independent set of size 4. (In fact, r(4, 4) = 18).

Caicedo Ramsey theory and small countable ordinals

In the course of an examination of friendship between children some fifty years ago, the Hungarian sociologist Sandor Szalai observed that among any group of about twenty children he checked, he could always find four children any two of whom were friends, or else four children no two of whom were friends. Despite the temptation to try to draw sociological conclusions, Szalai realized that this might well be a mathematical phenomenon rather than a sociological

• ne. Indeed, a brief discussion with the mathematicians

Erd˝

• s, Tur´

an, and S´

• s convinced him this was the case.

What Szalai is observing, in the language of Ramsey theory, is that r(4, 4) ≤ 20, that is, any graph on 20 vertices either contains a copy of K4, the complete graph on 4 vertices, or a copy of ¯ K4, the independent set of size 4. (In fact, r(4, 4) = 18).

Caicedo Ramsey theory and small countable ordinals

In the course of an examination of friendship between children some fifty years ago, the Hungarian sociologist Sandor Szalai observed that among any group of about twenty children he checked, he could always find four children any two of whom were friends, or else four children no two of whom were friends. Despite the temptation to try to draw sociological conclusions, Szalai realized that this might well be a mathematical phenomenon rather than a sociological

• ne. Indeed, a brief discussion with the mathematicians

Erd˝

• s, Tur´

an, and S´

• s convinced him this was the case.

What Szalai is observing, in the language of Ramsey theory, is that r(4, 4) ≤ 20, that is, any graph on 20 vertices either contains a copy of K4, the complete graph on 4 vertices, or a copy of ¯ K4, the independent set of size 4. (In fact, r(4, 4) = 18).

Caicedo Ramsey theory and small countable ordinals

We denote by n → (m, l)2 the statement that any graph on n vertices has a copy of Km or of ¯ Kl. This is equivalent to saying that whenever the edges of Kn are colored red and blue, either we have a red copy of Km, or a blue copy of Kl. We say that the relevant subgraph is monochromatic. (The superindex 2 indicates we are discussing edges, sets of size 2. There are corresponding statements for “triangles” or, in general, sets of size n for any n, but we are only concerned here with the case n = 2.)

Caicedo Ramsey theory and small countable ordinals

Theorem (Ramsey) For all m, l there is an n such that any graph on n vertices contains a copy of Km or ¯ Kl, that is, n → (m, l)2. Define the Ramsey number r(m, l) as the smallest possible value

• f n.

Clearly, r(n, m) = r(m, n), r(1, m) = 1, and r(2, m) = m.

Caicedo Ramsey theory and small countable ordinals

Frank Plumpton Ramsey (22 Feb., 1903 – 19 Jan., 1930). Philosopher, economist, mathematician. (Photograph by Lettice Ramsey, 1925.)

Caicedo Ramsey theory and small countable ordinals

For example: r(3, 3) = 6. To show that r(3, 3) is at least 6, it suffices to consider the following coloring of K5:

Caicedo Ramsey theory and small countable ordinals

r(3, 3) = 6. To see that it is at most 6, note that in any coloring of K6 there must be three edges with the same color and sharing a common vertex:

Caicedo Ramsey theory and small countable ordinals

r(4, 3) = 9. To see that it is at least 9, it suffices to consider the following graph: (A graph on 8 vertices without triangles of copies of ¯ K4.)

Caicedo Ramsey theory and small countable ordinals

r(4, 3) = 9. To see that r(4, 3) ≤ 10, consider the following diagram: (Ramsey theory, Graham-Rothschild-Spencer, p.4, c John Wiley & Sons, Inc.) The idea here is to generalize the argument used to prove that r(3, 3) ≤ 6. The general result is as follows:

Caicedo Ramsey theory and small countable ordinals

Lemma For all positive integers n, m, we have that r(n + 1, m + 1) ≤ r(n + 1, m) + r(n, m + 1). Arguing by induction, this lemma shows in particular that r(n, m) exists for all n, m. Since r(2, 3) = 3, for r(3, 3) this simply says that r(3, 3) ≤ 6. In the case at hand, it gives us that r(4, 3) ≤ r(4, 2) + r(3, 3) = 4 + 6 = 10.

Caicedo Ramsey theory and small countable ordinals

Proof. Let a = r(n + 1, m) and b = r(n, m + 1). Let’s use the metaphor suggested by Szalai’s anecdote. Suppose we have a group of a + b children, one of them named Ana. Either Ana is not friends with at least a other kids, or is friends with at least b of them. In the first case, since a = r(n + 1, m), either there are n + 1 among these a kids that are friends with one another, or else there are m, not 2 of whom are friends, in which case these m kids together with Ana form m + 1 friends, no 2 of whom are friends. Either way, we are done. The argument in the second case is analogous.

Caicedo Ramsey theory and small countable ordinals

Proof. Let a = r(n + 1, m) and b = r(n, m + 1). Let’s use the metaphor suggested by Szalai’s anecdote. Suppose we have a group of a + b children, one of them named Ana. Either Ana is not friends with at least a other kids, or is friends with at least b of them. In the first case, since a = r(n + 1, m), either there are n + 1 among these a kids that are friends with one another, or else there are m, not 2 of whom are friends, in which case these m kids together with Ana form m + 1 friends, no 2 of whom are friends. Either way, we are done. The argument in the second case is analogous.

Caicedo Ramsey theory and small countable ordinals

Proof. Let a = r(n + 1, m) and b = r(n, m + 1). Let’s use the metaphor suggested by Szalai’s anecdote. Suppose we have a group of a + b children, one of them named Ana. Either Ana is not friends with at least a other kids, or is friends with at least b of them. In the first case, since a = r(n + 1, m), either there are n + 1 among these a kids that are friends with one another, or else there are m, not 2 of whom are friends, in which case these m kids together with Ana form m + 1 friends, no 2 of whom are friends. Either way, we are done. The argument in the second case is analogous.

Caicedo Ramsey theory and small countable ordinals

Proof. Let a = r(n + 1, m) and b = r(n, m + 1). Let’s use the metaphor suggested by Szalai’s anecdote. Suppose we have a group of a + b children, one of them named Ana. Either Ana is not friends with at least a other kids, or is friends with at least b of them. In the first case, since a = r(n + 1, m), either there are n + 1 among these a kids that are friends with one another, or else there are m, not 2 of whom are friends, in which case these m kids together with Ana form m + 1 friends, no 2 of whom are friends. Either way, we are done. The argument in the second case is analogous.

Caicedo Ramsey theory and small countable ordinals

Proof. Let a = r(n + 1, m) and b = r(n, m + 1). Let’s use the metaphor suggested by Szalai’s anecdote. Suppose we have a group of a + b children, one of them named Ana. Either Ana is not friends with at least a other kids, or is friends with at least b of them. In the first case, since a = r(n + 1, m), either there are n + 1 among these a kids that are friends with one another, or else there are m, not 2 of whom are friends, in which case these m kids together with Ana form m + 1 friends, no 2 of whom are friends. Either way, we are done. The argument in the second case is analogous.

Caicedo Ramsey theory and small countable ordinals

Proof. Let a = r(n + 1, m) and b = r(n, m + 1). Let’s use the metaphor suggested by Szalai’s anecdote. Suppose we have a group of a + b children, one of them named Ana. Either Ana is not friends with at least a other kids, or is friends with at least b of them. In the first case, since a = r(n + 1, m), either there are n + 1 among these a kids that are friends with one another, or else there are m, not 2 of whom are friends, in which case these m kids together with Ana form m + 1 friends, no 2 of whom are friends. Either way, we are done. The argument in the second case is analogous.

Caicedo Ramsey theory and small countable ordinals

(To see that r(4, 3) is in fact at most 9 requires an additional argument.)

Caicedo Ramsey theory and small countable ordinals

r(4, 4) = 18. To see that it is at least 18, it suffices to consider the following graph: Coloring of K17 without monochromatic copies of K4.

Caicedo Ramsey theory and small countable ordinals

(This is the graph on Z/17Z where an edge {i, j} is colored blue if and only if i − j is ±2k for some k = 0, 1, 2, 3.) To see that r(4, 4) ≤ 18 it suffices to use the lemma.

Caicedo Ramsey theory and small countable ordinals

(This is the graph on Z/17Z where an edge {i, j} is colored blue if and only if i − j is ±2k for some k = 0, 1, 2, 3.) To see that r(4, 4) ≤ 18 it suffices to use the lemma.

Caicedo Ramsey theory and small countable ordinals

r(3, 5) = 14, r(3, 6) = 18, r(3, 7) = 23, r(3, 8) = 28, r(3, 9) = 36. 40 ≤ r(3, 10) ≤ 42. r(4, 5) = 25, 36 ≤ r(4, 6) ≤ 41. These results require extensive computations. For instance, that r(4, 5) ≥ 25 was shown by James Kalbfleisch in 1965. That r(4, 5) ≤ 25 was shown by Brendan McKay and Stanis law Radziszowski in 1993.

Caicedo Ramsey theory and small countable ordinals

r(3, 5) = 14, r(3, 6) = 18, r(3, 7) = 23, r(3, 8) = 28, r(3, 9) = 36. 40 ≤ r(3, 10) ≤ 42. r(4, 5) = 25, 36 ≤ r(4, 6) ≤ 41. These results require extensive computations. For instance, that r(4, 5) ≥ 25 was shown by James Kalbfleisch in 1965. That r(4, 5) ≤ 25 was shown by Brendan McKay and Stanis law Radziszowski in 1993.

Caicedo Ramsey theory and small countable ordinals

r(3, 5) = 14, r(3, 6) = 18, r(3, 7) = 23, r(3, 8) = 28, r(3, 9) = 36. 40 ≤ r(3, 10) ≤ 42. r(4, 5) = 25, 36 ≤ r(4, 6) ≤ 41. These results require extensive computations. For instance, that r(4, 5) ≥ 25 was shown by James Kalbfleisch in 1965. That r(4, 5) ≤ 25 was shown by Brendan McKay and Stanis law Radziszowski in 1993.

Caicedo Ramsey theory and small countable ordinals

r(3, 5) = 14, r(3, 6) = 18, r(3, 7) = 23, r(3, 8) = 28, r(3, 9) = 36. 40 ≤ r(3, 10) ≤ 42. r(4, 5) = 25, 36 ≤ r(4, 6) ≤ 41. These results require extensive computations. For instance, that r(4, 5) ≥ 25 was shown by James Kalbfleisch in 1965. That r(4, 5) ≤ 25 was shown by Brendan McKay and Stanis law Radziszowski in 1993.

Caicedo Ramsey theory and small countable ordinals

The two implementations required 3.2 years and 6 years

• f cpu time on Sun Microsystems computers (mostly

Sparcstation SLC). This was achieved without undue delay by employing a large number of computers (up to 110 at once). McKay-Radziszowski (1993)

Caicedo Ramsey theory and small countable ordinals

43 ≤ r(5, 5) ≤ 48. The lower bound is due to Geoffrey Exoo in 1989. The upper bound by Vigleik Angeltveit and McKay was posted on the ArXiv on March 26, 2017. In their paper, they explain that the proof is via computer verification, checking approximately two trillion separate cases. 102 ≤ r(6, 6) ≤ 165.

Caicedo Ramsey theory and small countable ordinals

43 ≤ r(5, 5) ≤ 48. The lower bound is due to Geoffrey Exoo in 1989. The upper bound by Vigleik Angeltveit and McKay was posted on the ArXiv on March 26, 2017. In their paper, they explain that the proof is via computer verification, checking approximately two trillion separate cases. 102 ≤ r(6, 6) ≤ 165.

Caicedo Ramsey theory and small countable ordinals

43 ≤ r(5, 5) ≤ 48. The lower bound is due to Geoffrey Exoo in 1989. The upper bound by Vigleik Angeltveit and McKay was posted on the ArXiv on March 26, 2017. In their paper, they explain that the proof is via computer verification, checking approximately two trillion separate cases. 102 ≤ r(6, 6) ≤ 165.

Caicedo Ramsey theory and small countable ordinals

43 ≤ r(5, 5) ≤ 48. The lower bound is due to Geoffrey Exoo in 1989. The upper bound by Vigleik Angeltveit and McKay was posted on the ArXiv on March 26, 2017. In their paper, they explain that the proof is via computer verification, checking approximately two trillion separate cases. 102 ≤ r(6, 6) ≤ 165.

Caicedo Ramsey theory and small countable ordinals

It must seem incredible to the uninitiated that in the age

• f supercomputers r2(5, 5) is unknown. This, of course,

is caused by the so-called combinatorial explosion: there are just too many cases to be checked. In fact, I made the following joke. Suppose an evil spirit would tell us: unless you give me the value of r2(5, 5) in a year, I will exterminate the human race. Our best strategy probably would be to have our computers working on r2(5, 5) and we could have the value of r2(5, 5) in time. If he would ask for r2(6, 6) our best strategy would be to try to destroy him/her/it before he destroys us. If we could give the answer by mathematics, we would be so clever and powerful that we could ignore it and tell it if you try to attack us you will see what will happen to you.

• P. Erd˝
• s (1993)

Caicedo Ramsey theory and small countable ordinals

Paul Erd˝

• s (26 Mar. 1913, 20 Sep. 1996). The picture, c

George Csicsery, was taken an the beginning of the Cambridge Combinatorial Conference in Honour of Paul Erd˝

• s (March 21–25,

1988), on occasion of his 75-th birthday.

Caicedo Ramsey theory and small countable ordinals

Ramsey’s theorem also admits infinitary versions, and in this context we talk of the partition calculus.

Caicedo Ramsey theory and small countable ordinals

Theorem (Ramsey) N → (N, N)2, that is, any graph on infinitely many vertices contains either a complete infinite graph, or an infinite independent graph.

Caicedo Ramsey theory and small countable ordinals

Proof. Here us a quick proof. Consider a red-blue coloring c of the edges

• f the complete graph on N. Let a0 = 0.

One of Aa00 = {i ∈ N | i > a0 and c({a0, i}) = red} and Aa01 = {i ∈ N | i > a0 and c({a0, i}) = blue} is infinite. Say it is

• Aa00. Let a1 = min(Aa00).

One of Aa10 = {i ∈ Aa00 | i > a1 and c({a1, i}) = red} and Aa11 = {i ∈ Aa00 | i > a1 and c({a1, i}) = blue} is infinite. Say it is Aa11. Let a2 = min(Aa11). One of Aa20 = {i ∈ Aa11 | i > a2 and c({a2, i}) = red} and Aa21 = {i ∈ Aa21 | i > a2 and c({a2, i}) = blue} is infinite. Say it is Aa21. Let a3 = min(Aa21). Etc. The point is that if i < j, k then c({ai, aj}) = c({ai, ak}). For each i, this color is either red or blue, so for infinitely many i it is the same. The subsequence consisting of these ai gives us an infinite monochromatic set.

Caicedo Ramsey theory and small countable ordinals

Proof. Here us a quick proof. Consider a red-blue coloring c of the edges

• f the complete graph on N. Let a0 = 0.

One of Aa00 = {i ∈ N | i > a0 and c({a0, i}) = red} and Aa01 = {i ∈ N | i > a0 and c({a0, i}) = blue} is infinite. Say it is

• Aa00. Let a1 = min(Aa00).

One of Aa10 = {i ∈ Aa00 | i > a1 and c({a1, i}) = red} and Aa11 = {i ∈ Aa00 | i > a1 and c({a1, i}) = blue} is infinite. Say it is Aa11. Let a2 = min(Aa11). One of Aa20 = {i ∈ Aa11 | i > a2 and c({a2, i}) = red} and Aa21 = {i ∈ Aa21 | i > a2 and c({a2, i}) = blue} is infinite. Say it is Aa21. Let a3 = min(Aa21). Etc. The point is that if i < j, k then c({ai, aj}) = c({ai, ak}). For each i, this color is either red or blue, so for infinitely many i it is the same. The subsequence consisting of these ai gives us an infinite monochromatic set.

Caicedo Ramsey theory and small countable ordinals

Proof. Here us a quick proof. Consider a red-blue coloring c of the edges

• f the complete graph on N. Let a0 = 0.

One of Aa00 = {i ∈ N | i > a0 and c({a0, i}) = red} and Aa01 = {i ∈ N | i > a0 and c({a0, i}) = blue} is infinite. Say it is

• Aa00. Let a1 = min(Aa00).

One of Aa10 = {i ∈ Aa00 | i > a1 and c({a1, i}) = red} and Aa11 = {i ∈ Aa00 | i > a1 and c({a1, i}) = blue} is infinite. Say it is Aa11. Let a2 = min(Aa11). One of Aa20 = {i ∈ Aa11 | i > a2 and c({a2, i}) = red} and Aa21 = {i ∈ Aa21 | i > a2 and c({a2, i}) = blue} is infinite. Say it is Aa21. Let a3 = min(Aa21). Etc. The point is that if i < j, k then c({ai, aj}) = c({ai, ak}). For each i, this color is either red or blue, so for infinitely many i it is the same. The subsequence consisting of these ai gives us an infinite monochromatic set.

Caicedo Ramsey theory and small countable ordinals

Proof. Here us a quick proof. Consider a red-blue coloring c of the edges

• f the complete graph on N. Let a0 = 0.

One of Aa00 = {i ∈ N | i > a0 and c({a0, i}) = red} and Aa01 = {i ∈ N | i > a0 and c({a0, i}) = blue} is infinite. Say it is

• Aa00. Let a1 = min(Aa00).

One of Aa10 = {i ∈ Aa00 | i > a1 and c({a1, i}) = red} and Aa11 = {i ∈ Aa00 | i > a1 and c({a1, i}) = blue} is infinite. Say it is Aa11. Let a2 = min(Aa11). One of Aa20 = {i ∈ Aa11 | i > a2 and c({a2, i}) = red} and Aa21 = {i ∈ Aa21 | i > a2 and c({a2, i}) = blue} is infinite. Say it is Aa21. Let a3 = min(Aa21). Etc. The point is that if i < j, k then c({ai, aj}) = c({ai, ak}). For each i, this color is either red or blue, so for infinitely many i it is the same. The subsequence consisting of these ai gives us an infinite monochromatic set.

Caicedo Ramsey theory and small countable ordinals

Proof. Here us a quick proof. Consider a red-blue coloring c of the edges

• f the complete graph on N. Let a0 = 0.

One of Aa00 = {i ∈ N | i > a0 and c({a0, i}) = red} and Aa01 = {i ∈ N | i > a0 and c({a0, i}) = blue} is infinite. Say it is

• Aa00. Let a1 = min(Aa00).

One of Aa10 = {i ∈ Aa00 | i > a1 and c({a1, i}) = red} and Aa11 = {i ∈ Aa00 | i > a1 and c({a1, i}) = blue} is infinite. Say it is Aa11. Let a2 = min(Aa11). One of Aa20 = {i ∈ Aa11 | i > a2 and c({a2, i}) = red} and Aa21 = {i ∈ Aa21 | i > a2 and c({a2, i}) = blue} is infinite. Say it is Aa21. Let a3 = min(Aa21). Etc. The point is that if i < j, k then c({ai, aj}) = c({ai, ak}). For each i, this color is either red or blue, so for infinitely many i it is the same. The subsequence consisting of these ai gives us an infinite monochromatic set.

Caicedo Ramsey theory and small countable ordinals

Proof. Here us a quick proof. Consider a red-blue coloring c of the edges

• f the complete graph on N. Let a0 = 0.

One of Aa00 = {i ∈ N | i > a0 and c({a0, i}) = red} and Aa01 = {i ∈ N | i > a0 and c({a0, i}) = blue} is infinite. Say it is

• Aa00. Let a1 = min(Aa00).

One of Aa10 = {i ∈ Aa00 | i > a1 and c({a1, i}) = red} and Aa11 = {i ∈ Aa00 | i > a1 and c({a1, i}) = blue} is infinite. Say it is Aa11. Let a2 = min(Aa11). One of Aa20 = {i ∈ Aa11 | i > a2 and c({a2, i}) = red} and Aa21 = {i ∈ Aa21 | i > a2 and c({a2, i}) = blue} is infinite. Say it is Aa21. Let a3 = min(Aa21). Etc. The point is that if i < j, k then c({ai, aj}) = c({ai, ak}). For each i, this color is either red or blue, so for infinitely many i it is the same. The subsequence consisting of these ai gives us an infinite monochromatic set.

Caicedo Ramsey theory and small countable ordinals

Countable ordinals

Many extensions are possible. Here I will consider, rather than just sets of vertices, ordered sets. And I further restrict to the case of (infinite, but rather small) ordinals. The ordinals are the order types of well-ordered sets. A linear order (X, <) is a well-order if and only if every non-empty subset of X has a first element. The usual order of the natural numbers is an example. Any two ordinals can be compared (one is an initial segment of the

• ther). What we obtain with the ordinals is a transfinite way of

continuing the sequence of the naturals.

Caicedo Ramsey theory and small countable ordinals

We have two operations at our disposal to continue the sequence: Given an ordinal α, we can “add 1” and obtain the successor

• rdinal α + 1.

Or we can consider a non-empty collection of ordinals, without maximum, and add its supremum. The ordinals so

• btained are the limit ordinals.

We denote by ω the first infinite ordinal: We begin with the naturals, 0, 1, 2, . . . , and then ω, ω + 1, ω + 2, . . . , ω2, ω2 + 1, . . . , ω3, . . . , ω2, . . . , ωω, . . .

Caicedo Ramsey theory and small countable ordinals

We have two operations at our disposal to continue the sequence: Given an ordinal α, we can “add 1” and obtain the successor

• rdinal α + 1.

Or we can consider a non-empty collection of ordinals, without maximum, and add its supremum. The ordinals so

• btained are the limit ordinals.

We denote by ω the first infinite ordinal: We begin with the naturals, 0, 1, 2, . . . , and then ω, ω + 1, ω + 2, . . . , ω2, ω2 + 1, . . . , ω3, . . . , ω2, . . . , ωω, . . .

Caicedo Ramsey theory and small countable ordinals

We have two operations at our disposal to continue the sequence: Given an ordinal α, we can “add 1” and obtain the successor

• rdinal α + 1.

Or we can consider a non-empty collection of ordinals, without maximum, and add its supremum. The ordinals so

• btained are the limit ordinals.

We denote by ω the first infinite ordinal: We begin with the naturals, 0, 1, 2, . . . , and then ω, ω + 1, ω + 2, . . . , ω2, ω2 + 1, . . . , ω3, . . . , ω2, . . . , ωω, . . .

Caicedo Ramsey theory and small countable ordinals

We have two operations at our disposal to continue the sequence: Given an ordinal α, we can “add 1” and obtain the successor

• rdinal α + 1.

Or we can consider a non-empty collection of ordinals, without maximum, and add its supremum. The ordinals so

• btained are the limit ordinals.

We denote by ω the first infinite ordinal: We begin with the naturals, 0, 1, 2, . . . , and then ω, ω + 1, ω + 2, . . . , ω2, ω2 + 1, . . . , ω3, . . . , ω2, . . . , ωω, . . .

Caicedo Ramsey theory and small countable ordinals

We have two operations at our disposal to continue the sequence: Given an ordinal α, we can “add 1” and obtain the successor

• rdinal α + 1.

Or we can consider a non-empty collection of ordinals, without maximum, and add its supremum. The ordinals so

• btained are the limit ordinals.

We denote by ω the first infinite ordinal: We begin with the naturals, 0, 1, 2, . . . , and then ω, ω + 1, ω + 2, . . . , ω2, ω2 + 1, . . . , ω3, . . . , ω2, . . . , ωω, . . .

Caicedo Ramsey theory and small countable ordinals

It is standard to identify each ordinal with its set of predecessors, so that α = [0, α). Definition If α, β, γ are ordinals, we say that α → (β, γ)2 if and only if for any red-blue coloring of the edges of Kα we can find a subset H ⊆ α with KH monochromatic and such that, if its color is red, then H has order type at least β while, if it is blue, then H has

• rder type at least γ.

As before, r(β, γ) is the least α such that α → (β, γ)2.

Caicedo Ramsey theory and small countable ordinals

For example, r(ω, ω) = ω (Ramsey), but already r(ω + 1, ω) = ω1, the first uncountable ordinal (Erd˝

• s-Rado, Ernst Specker).

But this means that if we want to restrict ourselves to countable resources, then we must necessarily accept that one of the ordinals under consideration is actually finite. On the other hand, Erd˝

• s and Eric Milner proved that if α is

countable, and n is finite, then r(α, n) is also countable.

Caicedo Ramsey theory and small countable ordinals

For example, r(ω, ω) = ω (Ramsey), but already r(ω + 1, ω) = ω1, the first uncountable ordinal (Erd˝

• s-Rado, Ernst Specker).

But this means that if we want to restrict ourselves to countable resources, then we must necessarily accept that one of the ordinals under consideration is actually finite. On the other hand, Erd˝

• s and Eric Milner proved that if α is

countable, and n is finite, then r(α, n) is also countable.

Caicedo Ramsey theory and small countable ordinals

For example, r(ω, ω) = ω (Ramsey), but already r(ω + 1, ω) = ω1, the first uncountable ordinal (Erd˝

• s-Rado, Ernst Specker).

But this means that if we want to restrict ourselves to countable resources, then we must necessarily accept that one of the ordinals under consideration is actually finite. On the other hand, Erd˝

• s and Eric Milner proved that if α is

countable, and n is finite, then r(α, n) is also countable.

Caicedo Ramsey theory and small countable ordinals

Can we be more precise? Consider an example: r(ω + 1, 3) = ω2 + 1. ω2 is not enough: It suffices to color red all edges in either copy of ω, and color blue all edges between both copies. As the diagram suggests, we are basically considering a graph on two vertices. [0, ω) [ω, ω2)

Caicedo Ramsey theory and small countable ordinals

Can we be more precise? Consider an example: r(ω + 1, 3) = ω2 + 1. ω2 is not enough: It suffices to color red all edges in either copy of ω, and color blue all edges between both copies. As the diagram suggests, we are basically considering a graph on two vertices. [0, ω) [ω, ω2)

Caicedo Ramsey theory and small countable ordinals

Can we be more precise? Consider an example: r(ω + 1, 3) = ω2 + 1. ω2 is not enough: It suffices to color red all edges in either copy of ω, and color blue all edges between both copies. As the diagram suggests, we are basically considering a graph on two vertices. [0, ω) [ω, ω2)

Caicedo Ramsey theory and small countable ordinals

Can we be more precise? Consider an example: r(ω + 1, 3) = ω2 + 1. ω2 is not enough: It suffices to color red all edges in either copy of ω, and color blue all edges between both copies. As the diagram suggests, we are basically considering a graph on two vertices. [0, ω) [ω, ω2)

Caicedo Ramsey theory and small countable ordinals

r(ω + 1, 3) = ω2 + 1. To see that ω2 + 1 is an upper bound, consider a coloring of Kω2+1. Since ω → (ω, ω)2, we may assume that all edges in K[0,ω) and K[ω,ω2) are red. If some α > ω is connected to infinitely many vertices in [0, ω) with a red edge, we are done. Essentially, this means we have reproduced the previous diagram. The (dotted) blue edge means that every point in [ω, ω2) is connected to almost every point in [0, ω) with a blue edge. [0, ω) [ω, ω2) But we have not considered yet the last point.

Caicedo Ramsey theory and small countable ordinals

r(ω + 1, 3) = ω2 + 1. To see that ω2 + 1 is an upper bound, consider a coloring of Kω2+1. Since ω → (ω, ω)2, we may assume that all edges in K[0,ω) and K[ω,ω2) are red. If some α > ω is connected to infinitely many vertices in [0, ω) with a red edge, we are done. Essentially, this means we have reproduced the previous diagram. The (dotted) blue edge means that every point in [ω, ω2) is connected to almost every point in [0, ω) with a blue edge. [0, ω) [ω, ω2) But we have not considered yet the last point.

Caicedo Ramsey theory and small countable ordinals

r(ω + 1, 3) = ω2 + 1. To see that ω2 + 1 is an upper bound, consider a coloring of Kω2+1. Since ω → (ω, ω)2, we may assume that all edges in K[0,ω) and K[ω,ω2) are red. If some α > ω is connected to infinitely many vertices in [0, ω) with a red edge, we are done. Essentially, this means we have reproduced the previous diagram. The (dotted) blue edge means that every point in [ω, ω2) is connected to almost every point in [0, ω) with a blue edge. [0, ω) [ω, ω2) But we have not considered yet the last point.

Caicedo Ramsey theory and small countable ordinals

r(ω + 1, 3) = ω2 + 1. To see that ω2 + 1 is an upper bound, consider a coloring of Kω2+1. Since ω → (ω, ω)2, we may assume that all edges in K[0,ω) and K[ω,ω2) are red. If some α > ω is connected to infinitely many vertices in [0, ω) with a red edge, we are done. Essentially, this means we have reproduced the previous diagram. The (dotted) blue edge means that every point in [ω, ω2) is connected to almost every point in [0, ω) with a blue edge. [0, ω) [ω, ω2) But we have not considered yet the last point.

Caicedo Ramsey theory and small countable ordinals

r(ω + 1, 3) = ω2 + 1. [0, ω) [ω, ω2) If the last point is connected to one of the big dots with a red edge, we are done (we have a red copy of Kω+1). If the edges are blue we are done as well (now we have a blue copy of K3): [0, ω) [ω, ω2) The point is that we have reduced the infinitary problem of identifying r(ω +1, 3) to a problem about colorings of finite graphs.

Caicedo Ramsey theory and small countable ordinals

r(ω + 1, 3) = ω2 + 1. [0, ω) [ω, ω2) If the last point is connected to one of the big dots with a red edge, we are done (we have a red copy of Kω+1). If the edges are blue we are done as well (now we have a blue copy of K3): [0, ω) [ω, ω2) The point is that we have reduced the infinitary problem of identifying r(ω +1, 3) to a problem about colorings of finite graphs.

Caicedo Ramsey theory and small countable ordinals

r(ω + 1, 3) = ω2 + 1. [0, ω) [ω, ω2) If the last point is connected to one of the big dots with a red edge, we are done (we have a red copy of Kω+1). If the edges are blue we are done as well (now we have a blue copy of K3): [0, ω) [ω, ω2) The point is that we have reduced the infinitary problem of identifying r(ω +1, 3) to a problem about colorings of finite graphs.

Caicedo Ramsey theory and small countable ordinals

The same ideas allow us to show, for example: r(ω + 2, 3) = ω2 + 4. For instance, the picture below verifies that r(ω + 2, 3) > ω2 + 3: [0, ω) [ω, ω2)

Caicedo Ramsey theory and small countable ordinals

The same ideas allow us to show, for example: r(ω + 2, 3) = ω2 + 4. For instance, the picture below verifies that r(ω + 2, 3) > ω2 + 3: [0, ω) [ω, ω2)

Caicedo Ramsey theory and small countable ordinals

r(ω + 2, 4) = ω3 + 7. r(ω + 3, 4) = ω3 + 16. In each case, lifting and projecting operations allow us to identify colorings of infinite graphs with colorings of finite graphs, reducing the computation of the ordinals r(ω + n, m) to similar problems to the computation of finite Ramsey numbers.

Caicedo Ramsey theory and small countable ordinals

r(ω + 2, 4) = ω3 + 7. r(ω + 3, 4) = ω3 + 16. In each case, lifting and projecting operations allow us to identify colorings of infinite graphs with colorings of finite graphs, reducing the computation of the ordinals r(ω + n, m) to similar problems to the computation of finite Ramsey numbers.

Caicedo Ramsey theory and small countable ordinals

r(ω + 2, 4) = ω3 + 7. r(ω + 3, 4) = ω3 + 16. In each case, lifting and projecting operations allow us to identify colorings of infinite graphs with colorings of finite graphs, reducing the computation of the ordinals r(ω + n, m) to similar problems to the computation of finite Ramsey numbers.

Caicedo Ramsey theory and small countable ordinals

For example, the following finite graph verifies that ω3 + 15 → (ω + 3, 4). For the sake of clarity, only the blue edges are depicted. v0 : [0, ω) v1 : [ω, ω2) v2 : [ω2, ω3) v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17

Caicedo Ramsey theory and small countable ordinals

For example, the following finite graph verifies that ω3 + 15 → (ω + 3, 4). For the sake of clarity, only the blue edges are depicted. v0 : [0, ω) v1 : [ω, ω2) v2 : [ω2, ω3) v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17

Caicedo Ramsey theory and small countable ordinals

The previous graph is one of only five possible. They were identified in 2014 by Kyle Beserra, then an undergraduate student at Boise State University and currently a graduate student at the University of North Texas. Kyle Beserra. Photograph by Stephanie Potter, 2014.

Caicedo Ramsey theory and small countable ordinals

Haddad-Sabbagh

To describe the general result, we need a lemma. Lemma For all positive integers n, m there exists a positive integer k ≥ n, m such that any red-blue coloring of the edges of K[0,k), and such that K[0,m) is blue, admits a subset H ⊂ [0, k) with KH monochromatic, and

1 Either KH is blue, and |H| = m + 1, 2 Or KH is red, |H| = n + 1, and H contains a number in

[0, m). (If we remove the requirements that K[0,m) is blue and H ∩ [0, m) = ∅, this is Ramsey’s theorem.) Denote by rHS(n + 1, m + 1) the least number k witnessing the lemma. (HS is for Labib Haddad and Gabriel Sabbagh.)

Caicedo Ramsey theory and small countable ordinals

Haddad-Sabbagh

To describe the general result, we need a lemma. Lemma For all positive integers n, m there exists a positive integer k ≥ n, m such that any red-blue coloring of the edges of K[0,k), and such that K[0,m) is blue, admits a subset H ⊂ [0, k) with KH monochromatic, and

1 Either KH is blue, and |H| = m + 1, 2 Or KH is red, |H| = n + 1, and H contains a number in

[0, m). (If we remove the requirements that K[0,m) is blue and H ∩ [0, m) = ∅, this is Ramsey’s theorem.) Denote by rHS(n + 1, m + 1) the least number k witnessing the lemma. (HS is for Labib Haddad and Gabriel Sabbagh.)

Caicedo Ramsey theory and small countable ordinals

Haddad-Sabbagh

To describe the general result, we need a lemma. Lemma For all positive integers n, m there exists a positive integer k ≥ n, m such that any red-blue coloring of the edges of K[0,k), and such that K[0,m) is blue, admits a subset H ⊂ [0, k) with KH monochromatic, and

1 Either KH is blue, and |H| = m + 1, 2 Or KH is red, |H| = n + 1, and H contains a number in

[0, m). (If we remove the requirements that K[0,m) is blue and H ∩ [0, m) = ∅, this is Ramsey’s theorem.) Denote by rHS(n + 1, m + 1) the least number k witnessing the lemma. (HS is for Labib Haddad and Gabriel Sabbagh.)

Caicedo Ramsey theory and small countable ordinals

Haddad-Sabbagh

To describe the general result, we need a lemma. Lemma For all positive integers n, m there exists a positive integer k ≥ n, m such that any red-blue coloring of the edges of K[0,k), and such that K[0,m) is blue, admits a subset H ⊂ [0, k) with KH monochromatic, and

1 Either KH is blue, and |H| = m + 1, 2 Or KH is red, |H| = n + 1, and H contains a number in

[0, m). (If we remove the requirements that K[0,m) is blue and H ∩ [0, m) = ∅, this is Ramsey’s theorem.) Denote by rHS(n + 1, m + 1) the least number k witnessing the lemma. (HS is for Labib Haddad and Gabriel Sabbagh.)

Caicedo Ramsey theory and small countable ordinals

Haddad-Sabbagh

To describe the general result, we need a lemma. Lemma For all positive integers n, m there exists a positive integer k ≥ n, m such that any red-blue coloring of the edges of K[0,k), and such that K[0,m) is blue, admits a subset H ⊂ [0, k) with KH monochromatic, and

1 Either KH is blue, and |H| = m + 1, 2 Or KH is red, |H| = n + 1, and H contains a number in

[0, m). (If we remove the requirements that K[0,m) is blue and H ∩ [0, m) = ∅, this is Ramsey’s theorem.) Denote by rHS(n + 1, m + 1) the least number k witnessing the lemma. (HS is for Labib Haddad and Gabriel Sabbagh.)

Caicedo Ramsey theory and small countable ordinals

Haddad-Sabbagh

To describe the general result, we need a lemma. Lemma For all positive integers n, m there exists a positive integer k ≥ n, m such that any red-blue coloring of the edges of K[0,k), and such that K[0,m) is blue, admits a subset H ⊂ [0, k) with KH monochromatic, and

1 Either KH is blue, and |H| = m + 1, 2 Or KH is red, |H| = n + 1, and H contains a number in

[0, m). (If we remove the requirements that K[0,m) is blue and H ∩ [0, m) = ∅, this is Ramsey’s theorem.) Denote by rHS(n + 1, m + 1) the least number k witnessing the lemma. (HS is for Labib Haddad and Gabriel Sabbagh.)

Caicedo Ramsey theory and small countable ordinals

For example: rHS(2, m) = m. rHS(3, m) = m(m + 1) 2 . rHS(n + 1, 3) = r(n, 3) + n + 1. rHS(4, 3) = 10. rHS(4, 4) = 19. If m, n are positive integers, then rHS(n + 1, m + 1) ≤ r(n, m + 1) + rHS(n + 1, m).

Caicedo Ramsey theory and small countable ordinals

For example: rHS(2, m) = m. rHS(3, m) = m(m + 1) 2 . rHS(n + 1, 3) = r(n, 3) + n + 1. rHS(4, 3) = 10. rHS(4, 4) = 19. If m, n are positive integers, then rHS(n + 1, m + 1) ≤ r(n, m + 1) + rHS(n + 1, m).

Caicedo Ramsey theory and small countable ordinals

For example: rHS(2, m) = m. rHS(3, m) = m(m + 1) 2 . rHS(n + 1, 3) = r(n, 3) + n + 1. rHS(4, 3) = 10. rHS(4, 4) = 19. If m, n are positive integers, then rHS(n + 1, m + 1) ≤ r(n, m + 1) + rHS(n + 1, m).

Caicedo Ramsey theory and small countable ordinals

For example: rHS(2, m) = m. rHS(3, m) = m(m + 1) 2 . rHS(n + 1, 3) = r(n, 3) + n + 1. rHS(4, 3) = 10. rHS(4, 4) = 19. If m, n are positive integers, then rHS(n + 1, m + 1) ≤ r(n, m + 1) + rHS(n + 1, m).

Caicedo Ramsey theory and small countable ordinals

For example: rHS(2, m) = m. rHS(3, m) = m(m + 1) 2 . rHS(n + 1, 3) = r(n, 3) + n + 1. rHS(4, 3) = 10. rHS(4, 4) = 19. If m, n are positive integers, then rHS(n + 1, m + 1) ≤ r(n, m + 1) + rHS(n + 1, m).

Caicedo Ramsey theory and small countable ordinals

For example: rHS(2, m) = m. rHS(3, m) = m(m + 1) 2 . rHS(n + 1, 3) = r(n, 3) + n + 1. rHS(4, 3) = 10. rHS(4, 4) = 19. If m, n are positive integers, then rHS(n + 1, m + 1) ≤ r(n, m + 1) + rHS(n + 1, m).

Caicedo Ramsey theory and small countable ordinals

For example: rHS(2, m) = m. rHS(3, m) = m(m + 1) 2 . rHS(n + 1, 3) = r(n, 3) + n + 1. rHS(4, 3) = 10. rHS(4, 4) = 19. If m, n are positive integers, then rHS(n + 1, m + 1) ≤ r(n, m + 1) + rHS(n + 1, m).

Caicedo Ramsey theory and small countable ordinals

rHS(4, 4) = 19 = 3 + 16. The graphs verifying that rHS(4, 4) > 18 are precisely the graphs identified by Beserra.

Caicedo Ramsey theory and small countable ordinals

rHS(4, 4) = 19 = 3 + 16. The graphs verifying that rHS(4, 4) > 18 are precisely the graphs identified by Beserra.

Caicedo Ramsey theory and small countable ordinals

Theorem (Haddad-Sabbagh) Given positive integers n and m, we have that r(ω + n, m) = ω(m − 1) + t, where m − 1 + t = rHS(1 + n, m).

Caicedo Ramsey theory and small countable ordinals

Haddad and Sabbagh announced this result in Comptes Rendus in 1969, but never published the proof. As it was, the proof was written down, bare, with no comments or hints. Our note got very little attention, in fact it got almost none! So no further details were ever published.

• L. Haddad (2006)

Caicedo Ramsey theory and small countable ordinals

Although work in the area has continued, and many quite sophisticated results are known, more than 45 years later, no proof

• f the theorems of Haddad and Sabbagh has been published. I

have written a survey providing the details.

Caicedo Ramsey theory and small countable ordinals

Many thanks!

Caicedo Ramsey theory and small countable ordinals