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 one. Indeed, a brief discussion with the mathematicians Erd˝ os, Tur´ an, and S´ os 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 K 4 , the complete graph on 4 vertices, or a copy of ¯ K 4 , 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 one. Indeed, a brief discussion with the mathematicians Erd˝ os, Tur´ an, and S´ os 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 K 4 , the complete graph on 4 vertices, or a copy of ¯ K 4 , 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 one. Indeed, a brief discussion with the mathematicians Erd˝ os, Tur´ an, and S´ os 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 K 4 , the complete graph on 4 vertices, or a copy of ¯ K 4 , 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 K m or of ¯ K l . This is equivalent to saying that whenever the edges of K n are colored red and blue, either we have a red copy of K m , or a blue copy of K l . 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 K m or ¯ K l , that is, n → ( m, l ) 2 . Define the Ramsey number r ( m, l ) as the smallest possible value of 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 K 5 : 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 K 6 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 ¯ K 4 .) 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
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