On Three Sets with Nondecreasing Diameter Carl Yerger Davidson - - PowerPoint PPT Presentation

On Three Sets with Nondecreasing Diameter

Carl Yerger

Davidson College cayerger@davidson.edu Joint work with Daniel Bernstein, Davidson College and David Grynkiewicz, Karl-Franzens University of Graz

May 13, 2011

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Background - The Pigeonhole Principle

In 1961, Erd˝

• s, Ginzburg and Ziv proved the following theorem that is

the subject of many papers and generalizations.

Theorem

Let m ∈ N. Every sequence of 2m − 1 elements from Z contains a subsequence of m elements whose sum is zero modulo m. Notice that this theorem is a generalization of the pigeonhole

• principle. For instance, if the sequence contains only the residues 0

and 1, then this theorem describes the situation of placing 2m − 1 pigeons in 2 holes.

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Preliminaries

Some Definitions

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Preliminaries

Some Definitions

Let ∆ : X − → C, where C is the set of colors. This could be integers, residues, etc.

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Preliminaries

Some Definitions

Let ∆ : X − → C, where C is the set of colors. This could be integers, residues, etc. If C = {1, . . . , k}, ∆ is a k-coloring.

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Preliminaries

Some Definitions

Let ∆ : X − → C, where C is the set of colors. This could be integers, residues, etc. If C = {1, . . . , k}, ∆ is a k-coloring. If set C = Z, we call ∆ a Z-coloring.

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Preliminaries

Some Definitions

Let ∆ : X − → C, where C is the set of colors. This could be integers, residues, etc. If C = {1, . . . , k}, ∆ is a k-coloring. If set C = Z, we call ∆ a Z-coloring. A set X is called monochromatic if ∆(x) = ∆(x′) for all x, x′ ∈ X.

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Preliminaries

Some Definitions

Let ∆ : X − → C, where C is the set of colors. This could be integers, residues, etc. If C = {1, . . . , k}, ∆ is a k-coloring. If set C = Z, we call ∆ a Z-coloring. A set X is called monochromatic if ∆(x) = ∆(x′) for all x, x′ ∈ X. In a Z-coloring of X, a subset Y of X is called zero-sum modulo n if

• y∈Y ∆(y) ≡ 0 mod n.

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Preliminaries

Some Definitions

Let ∆ : X − → C, where C is the set of colors. This could be integers, residues, etc. If C = {1, . . . , k}, ∆ is a k-coloring. If set C = Z, we call ∆ a Z-coloring. A set X is called monochromatic if ∆(x) = ∆(x′) for all x, x′ ∈ X. In a Z-coloring of X, a subset Y of X is called zero-sum modulo n if

• y∈Y ∆(y) ≡ 0 mod n.

If a, b ∈ Z, then [a, b] is the set of integers {n ∈ Z|a ≤ n ≤ b}.

Carl Yerger (Davidson College) On Three Sets... May 13, 2011 3 / 18

Preliminaries

Some Definitions

Let ∆ : X − → C, where C is the set of colors. This could be integers, residues, etc. If C = {1, . . . , k}, ∆ is a k-coloring. If set C = Z, we call ∆ a Z-coloring. A set X is called monochromatic if ∆(x) = ∆(x′) for all x, x′ ∈ X. In a Z-coloring of X, a subset Y of X is called zero-sum modulo n if

• y∈Y ∆(y) ≡ 0 mod n.

If a, b ∈ Z, then [a, b] is the set of integers {n ∈ Z|a ≤ n ≤ b}. For finite X ⊆ N, define the diameter of X, denoted by diam(X), to be diam(X) = max(X) − min(X).

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A Coloring Problem

Coloring Set-up

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A Coloring Problem

Coloring Set-up

Let f (s, r, k) be the the smallest positive integer n such that for every coloring ∆ : [1, n] − → [1, k] there exist two subsets S1, S2 of [1, n], which satisfy:

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A Coloring Problem

Coloring Set-up

Let f (s, r, k) be the the smallest positive integer n such that for every coloring ∆ : [1, n] − → [1, k] there exist two subsets S1, S2 of [1, n], which satisfy: (a)S1 and S2 are monochromatic (b)|S1| = s, |S2| = r, (c) max(S1) < min(S2), and (d)diam(S1) ≤ diam(S2).

Theorem

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A Coloring Problem

Coloring Set-up

Let f (s, r, k) be the the smallest positive integer n such that for every coloring ∆ : [1, n] − → [1, k] there exist two subsets S1, S2 of [1, n], which satisfy: (a)S1 and S2 are monochromatic (b)|S1| = s, |S2| = r, (c) max(S1) < min(S2), and (d)diam(S1) ≤ diam(S2).

Theorem

In a paper of Bialostocki, Erd˝

• s and Lefmann (1995), it was shown

that f (m, m, 2) = 5m − 3 and f (m, m, 3) = 9m − 7.

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A More General Coloring Problem

Two Types of Problems (one in parentheses)

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A More General Coloring Problem

Two Types of Problems (one in parentheses)

Let f (s, r, Z)) ( f (s, r, Z) ∪ {∞})) be the the smallest positive integer n such that for every coloring ∆ : [1, n] − → Z (∆ : [1, n] − → {∞} ∪ Z), there exist two subsets S1, S2 of [1, n], which satisfy: (a)S1 is zero-sum mod s and S2 is zero-sum mod r (S1 is either ∞-monochromatic or zero-sum mod s and S2 is either ∞-monochromatic

• r zero-sum mod r),

(b)|S1| = s, |S2| = r, (c) max(S1) < min(S2), and (d)diam(S1) ≤ diam(S2).

Theorem

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A More General Coloring Problem

Two Types of Problems (one in parentheses)

Let f (s, r, Z)) ( f (s, r, Z) ∪ {∞})) be the the smallest positive integer n such that for every coloring ∆ : [1, n] − → Z (∆ : [1, n] − → {∞} ∪ Z), there exist two subsets S1, S2 of [1, n], which satisfy: (a)S1 is zero-sum mod s and S2 is zero-sum mod r (S1 is either ∞-monochromatic or zero-sum mod s and S2 is either ∞-monochromatic

• r zero-sum mod r),

(b)|S1| = s, |S2| = r, (c) max(S1) < min(S2), and (d)diam(S1) ≤ diam(S2).

Theorem

The paper of Bialostocki, Erd˝

• s and Lefmann, also shows that

f (m, m, 2) = f (m, m, Z) = 5m − 3 and f (m, m, 3) = f (m, m, {∞} ∪ Z) = 9m − 7. Such theorems are known as zero-sum generalizations in the sense of Erd˝

• s-Ginzburg-Ziv.

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Lower Bound Constructions

Lower bounds for 2-colorings and 3-colorings

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Lower Bound Constructions

Lower bounds for 2-colorings and 3-colorings

To show that f (m, m, 2) > 5m − 4 consider: 21m−12m−11m−122m−2

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Lower Bound Constructions

Lower bounds for 2-colorings and 3-colorings

To show that f (m, m, 2) > 5m − 4 consider: 21m−12m−11m−122m−2 To see that f (m, m, 3) > 9m − 8, consider: 31m−12m−13m−11m−12m−11m−12m−132m−2

Lower bounds for EGZ generalizations

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Lower Bound Constructions

Lower bounds for 2-colorings and 3-colorings

To show that f (m, m, 2) > 5m − 4 consider: 21m−12m−11m−122m−2 To see that f (m, m, 3) > 9m − 8, consider: 31m−12m−13m−11m−12m−11m−12m−132m−2

Lower bounds for EGZ generalizations

To show that f (m, m, Z) > 5m − 4 consider: 10m−11m−10m−112m−2

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Lower Bound Constructions

Lower bounds for 2-colorings and 3-colorings

To show that f (m, m, 2) > 5m − 4 consider: 21m−12m−11m−122m−2 To see that f (m, m, 3) > 9m − 8, consider: 31m−12m−13m−11m−12m−11m−12m−132m−2

Lower bounds for EGZ generalizations

To show that f (m, m, Z) > 5m − 4 consider: 10m−11m−10m−112m−2 To see that f (m, m, {∞} ∪ Z) > 9m − 8, consider: ∞0m−11m−1∞m−10m−11m−10m−11m−1∞2m−2

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Generalization of these Problems

What if the two sets are different sizes? (CY, 2005)

For r ≥ s ≥ 2, f (s, r, 2) =    5s − 3 if s = r 4s + r − 3 if s < r ≤ 2s − 2 2s + 2r − 2 if r > 2s − 2 For r ≥ s ≥ 3, f (s, r, Z) =    5s − 3 if s = r 4s + max(r, s +

s (r,s) − 1) − 3

if s < r ≤ 2s − 2 2s + 2r − 2 if r > 2s − 2 Note: for integers, s, r, let (s, r) be the greatest common factor of s and

• r. So (8, 12) = 4.

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Generalization of these Problems

What if the two sets are different sizes? (CY, 2005)

For r ≥ s ≥ 3, f (s, r, 3) =        9s − 7 if r ≤ ⌊5s−1

3 ⌋ − 1

4s + 3r − 4 if ⌊ 5s−1

3 ⌋ − 1 < r ≤ 2s − 2

6s + 2r − 6 if 2s − 2 < r ≤ 3s − 3 3s + 3r − 4 if r > 3s − 3 (r ≥ s ≥ 2) For r ≥ s ≥ 3, f (s, r, {∞} ∪ Z) =   

9s − 7 +

s (r,s) − 1

if r ≤ ⌊ 5s−1

3 ⌋ − 1

3s + 2r − 3 + min(3s − 3, s + r − 1 +

s (r,s) − 1)

if ⌊ 5s−1

3 ⌋ − 1 < r ≤ 3s − 3

3s + 3r − 4 if r > 3s − 3 (r ≥ s ≥ 2)

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Other Generalizations of this problem

Other Results

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Other Generalizations of this problem

Other Results

Bi Ji Wong worked on the opposite off-diagonal cases (i.e. if s ≥ r ≥ 2) and showed that this generalizes in the sense of EGZ.

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Other Generalizations of this problem

Other Results

Bi Ji Wong worked on the opposite off-diagonal cases (i.e. if s ≥ r ≥ 2) and showed that this generalizes in the sense of EGZ. Grynkiewicz showed in 2008 that f (m, m, 4) = 12m − 9 and that this case also zero-sum generalizes.

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Other Generalizations of this problem

Other Results

Bi Ji Wong worked on the opposite off-diagonal cases (i.e. if s ≥ r ≥ 2) and showed that this generalizes in the sense of EGZ. Grynkiewicz showed in 2008 that f (m, m, 4) = 12m − 9 and that this case also zero-sum generalizes. Bialostocki and Wilson looked at monochromatic sets of integers whose diameters form a monotone sequence.

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Other Generalizations of this problem

Other Results

Bi Ji Wong worked on the opposite off-diagonal cases (i.e. if s ≥ r ≥ 2) and showed that this generalizes in the sense of EGZ. Grynkiewicz showed in 2008 that f (m, m, 4) = 12m − 9 and that this case also zero-sum generalizes. Bialostocki and Wilson looked at monochromatic sets of integers whose diameters form a monotone sequence. They showed that for two monochromatic sets of size m at most 4m − 2 integers are needed, and for three monochromatic sets of size m, at most 8m − 5 integers are needed. They also give a general inequality for larger collections of monochromatic sets.

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Other Generalizations... continued

A Modified Diameter Problem

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Other Generalizations... continued

A Modified Diameter Problem

Another result of Grynkiewicz and Sabar looks at a modified diameter function (where the nondecreasing diameter condition only applies to the first j terms of each monochromatic set).

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Other Generalizations... continued

A Modified Diameter Problem

Another result of Grynkiewicz and Sabar looks at a modified diameter function (where the nondecreasing diameter condition only applies to the first j terms of each monochromatic set). They prove that fj(m, m, 2) ≤ 5m − 3, with equality holding for j = m.

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Other Generalizations... continued

A Modified Diameter Problem

Another result of Grynkiewicz and Sabar looks at a modified diameter function (where the nondecreasing diameter condition only applies to the first j terms of each monochromatic set). They prove that fj(m, m, 2) ≤ 5m − 3, with equality holding for j = m. Their paper involves the use of a strengthening of the Erd˝

• s-Ginzburg-Ziv theorem as well as a partition analogue of the

Cauchy-Davenport theorem.

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Our Result

The Set-up

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Our Result

The Set-up

Suppose that X <p Y if and only if max(X) < min(Y ).

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Our Result

The Set-up

Suppose that X <p Y if and only if max(X) < min(Y ). For a positive integer m, let f (m, m, m; 2) be the least integer N such that for every 2-coloring ∆ : [1, N] → {0, 1} there exist three subsets B1, B2, B3 ⊆ [1, N] such that

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Our Result

The Set-up

Suppose that X <p Y if and only if max(X) < min(Y ). For a positive integer m, let f (m, m, m; 2) be the least integer N such that for every 2-coloring ∆ : [1, N] → {0, 1} there exist three subsets B1, B2, B3 ⊆ [1, N] such that (a) Bi for i = 1, 2, 3 is monochromatic, (b) |Bi| = m for i = 1, 2, 3, (c) B1 <p B2 <p B3, and (d) diam(B1) ≤ diam(B2) ≤ diam(B3).

Theorem

We show that f (m, m, m; 2) = 8m − 5 + ⌊ 2m−2

3

⌋ for m ≥ 5.

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A General Strategy

How does the proof go?

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A General Strategy

How does the proof go?

We first investigate the structure of 2-colorings within the interval [1, 3m − 2].

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A General Strategy

How does the proof go?

We first investigate the structure of 2-colorings within the interval [1, 3m − 2]. We prove a structural lemma and then translate it to give a characterization of monochromatic m-sets.

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A General Strategy

How does the proof go?

We first investigate the structure of 2-colorings within the interval [1, 3m − 2]. We prove a structural lemma and then translate it to give a characterization of monochromatic m-sets. A lower bound is given via an explicit construction that avoids three monochromatic sets of nondecreasing diameter.

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A General Strategy

How does the proof go?

We first investigate the structure of 2-colorings within the interval [1, 3m − 2]. We prove a structural lemma and then translate it to give a characterization of monochromatic m-sets. A lower bound is given via an explicit construction that avoids three monochromatic sets of nondecreasing diameter. We then use the structural lemma to help us directly prove an upper bound.

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The Lower Bound

The Explicit Construction

Observe that f (m, m, m; 2) ≥ 8m − 5 + ⌊ 2m−2

3

⌋ follows by considering the coloring of [1, 8m − 6 + ⌊ 2m−2

3

⌋] given by the string 01m−10m−11m−10⌊ 2m−2

3

⌋1m−⌊ 2m−2

3

⌋−10m−112m−1+⌊ 2m−2

3

⌋0m−1.

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The Upper Bound

A Structural Lemma

Let ∆ : [1, 3m − 2] → {0, 1} be a 2-coloring such that there exists a monochromatic m-subset B with diam(B) ≥ 2m − 2, let 3m − 2 − β be the minimal integer such that there exists a monochromatic m-subset B with diam(B) ≥ 2m − 2, and let B1 be the monochromatic m-subset with diam(B1) minimal subject to max B1 = 3m − 2 − β and diam(B1) ≥ 2m − 2. Let diam(B1) = 2m − 2 + α, and suppose ∆(B1) = 1. Then one of the following holds. (i) ∆([1, 3m − 2 − β]) = 1m−β−1−νH00H11ν+1, where µ, ν ≥ 0 are integers and H1 is a string of length m − β − 2 + µ that contains exactly µ 1’s, and H0 is a string of length m + β − µ − 1 that, provided α > 1, contains exactly β − µ 1’s. Furthermore, either β = m − α − 1 or ν = µ = 0. ... and so on. There are three conditions.

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A useful lemma

The next lemma translates the structural information from the previous lemma into the existence of sets with small and medium size diameter.

Lemma

Let ∆ : [1, 3m − 2] → {0, 1} be a 2-coloring. (i) If there does not exist a monochromatic m-subset B such that diam(B) ≥ 2m − 2, then there exist monochromatic m-subsets D1, D2 ⊆ [1, 3m − 2], such that D1 <p D2, |D1| = |D2| = m, and diam(D1) = diam(D2) = m − 1. (ii) Otherwise, there exists a monochromatic m-subset A1 ⊆ [1, 3m − 2 − α − β] with diam(A1) ≤ 2m − 2 − α, and there exists a monochromatic m-subset A2 ⊆ [1, 3m − 2 − α − β] with diam(A2) ≤ m + ⌊ m+β−1

2

⌋ − 1. If α > 1, then diam(A1) ≤ m + β − 1. If the previous lemma holds and α > 1, then diam(A1) ≤ m + β − 1 − µ.

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Upper Bound Argument

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Upper Bound Argument

We use the pigeonhole principle to find a monochromatic set within the interval [−2m + 2, 0].

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Upper Bound Argument

We use the pigeonhole principle to find a monochromatic set within the interval [−2m + 2, 0]. This allows us to use the previously mentioned lemma on the interval [1, 3m − 2].

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Upper Bound Argument

We use the pigeonhole principle to find a monochromatic set within the interval [−2m + 2, 0]. This allows us to use the previously mentioned lemma on the interval [1, 3m − 2]. Essentially all of the work involves the case where we can find a monochromatic set B of diameter at least 2m − 2. The remainder of the argument looks to find a third monochromatic set with diameter at least the size of B.

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Further Directions

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Further Directions

Does this generalize in the sense of Erd˝

• s-Ginzburg-Ziv?

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Further Directions

Does this generalize in the sense of Erd˝

• s-Ginzburg-Ziv?

Can we simplify the proof by looking at the first set in more detail?

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Further Directions

Does this generalize in the sense of Erd˝

• s-Ginzburg-Ziv?

Can we simplify the proof by looking at the first set in more detail? Can we investigate other 3-set variants? There are now different sorts

• f off-diagonal cases.

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Thank you! Are there any questions?

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