Local Properties via Color Energy Graphs and Forbidden Configurations

Fish, Sara; Pohoata, Cosmin; Sheffer, Adam

doi:10.1137/18m1225987

Cited by 12 publications

(35 citation statements)

References 14 publications

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“…Since f 1 (x) is non-negative when x ≥ 0, we conclude that (5) holds. This completes the proof of (3), and thus the proof of the lemma.…”

Section: The Constructionmentioning

confidence: 78%

“…That is, the boundary between a trivial problem and a non-trivial one passes between ℓ ≥ k 2 − ⌊k/2⌋ + 2 and ℓ ≤ k 2 − ⌊k/2⌋ + 1. (Recently, a stronger lower bound for this case was derived by Fish, Pohoata, and Sheffer [5]. ) Fox, Pach, and Suk [6] proved that for every ε > 0,…”

Section: Introductionmentioning

confidence: 99%

“…The known upper bounds for g(n, k, ℓ) are significantly stronger than the equivalent bounds for φ(n, k, ℓ). In particular, Fish, Pohoata, and Sheffer [5] derive a family of bounds such as…”

Section: Introductionmentioning

confidence: 99%

“…Some works on local properties problems, such as [5,9], consider the difference set as containing only the positive differences. This change does not affect the problem, since the asymptotic size of A − A remains unchanged when moving from one definition to the other.…”

Section: Introductionmentioning

confidence: 99%

“…The current work relies on the standard definition of A − A, which should be kept in mind when comparing it to others. For example, the bound (2) looks slightly differently in [5] because of this difference of notation.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A construction for difference sets with local properties

Fish

Lund

Sheffer

2019

European Journal of Combinatorics

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We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set A of n real numbers such that |A − A| = n log 2 3 and that every subset A ′ ⊆ A of size k satisfies |A ′ − A ′ | ≥ k log 2 3 . This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties. *

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“…Since f 1 (x) is non-negative when x ≥ 0, we conclude that (5) holds. This completes the proof of (3), and thus the proof of the lemma.…”

Section: The Constructionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A construction for difference sets with local properties

Fish

Lund

Sheffer

2019

European Journal of Combinatorics

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The Erdős–Gyárfás function with respect to Gallai‐colorings

Broersma

Wang

2022

Journal of Graph Theory

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For fixed p $p$ and q $q$, an edge‐coloring of the complete graph K n ${K}_{n}$ is said to be a ( p , q ) $(p,q)$‐coloring if every K p ${K}_{p}$ receives at least q $q$ distinct colors. The function f ( n , p , q ) $f(n,p,q)$ is the minimum number of colors needed for K n ${K}_{n}$ to have a ( p , q ) $(p,q)$‐coloring. This function was introduced about 45 years ago, but was studied systematically by Erdős and Gyárfás in 1997, and is now known as the Erdős–Gyárfás function. In this paper, we study f ( n , p , q ) $f(n,p,q)$ with respect to Gallai‐colorings, where a Gallai‐coloring is an edge‐coloring of K n ${K}_{n}$ without rainbow triangles. Combining the two concepts, we consider the function g ( n , p , q ) $g(n,p,q)$ that is the minimum number of colors needed for a Gallai‐ ( p , q ) $(p,q)$‐coloring of K n ${K}_{n}$. Using the anti‐Ramsey number for K 3 ${K}_{3}$, we have that g ( n , p , q ) $g(n,p,q)$ is nontrivial only for 2 ≤ q ≤ p − 1 $2\le q\le p-1$. We give a general lower bound for this function and we study how this function falls off from being equal to n − 1 $n-1$ when q = p − 1 $q=p-1$ and p ≥ 4 $p\ge 4$ to being Θ ( log n ) ${\rm{\Theta }}(\mathrm{log}\unicode{x0200A}n)$ when q = 2 $q=2$. In particular, for appropriate p $p$ and n $n$, we prove that g = n − c $g=n-c$ when q = p − c $q=p-c$ and c ∈ MathClass-open{ 1 , 2 MathClass-close} $c\in \{1,2\}$, g $g$ is at most a fractional power of n $n$ when q = MathClass-open⌊ p − 1 MathClass-close⌋ $q=\lfloor \sqrt{p-1}\rfloor $, and g $g$ is logarithmic in n $n$ when 2 ≤ q ≤ MathClass-open⌊ log 2 ( p − 1 ) MathClass-close⌋ + 1 $2\le q\le \lfloor {\mathrm{log}}_{2}(p-1)\rfloor +1$.

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Lower bounds on the Erdős–Gyárfás problem via color energy graphs

Balogh

English

Heath

et al. 2023

Journal of Graph Theory

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Given positive integers p and q, a p q ( , )-coloring of the complete graph K n is an edge-coloring in which every pclique receives at least q colors. Erdős and Shelah posed the question of determining f n p q ( , , ), the minimum number of colors needed for a p q ( , )-coloring of K n . In this paper, we expand on the color energy technique introduced by Pohoata and Sheffer to prove new lower bounds on this function, making explicit the connection between bounds on extremal numbers and f n p q ( , , ).Using results on the extremal numbers of subdivided complete graphs, theta graphs, and subdivided complete bipartite graphs, we generalize results of Fish, Pohoata, and Sheffer, giving the first nontrivial lower bounds on f n p q ( , , ) for some pairs p q ( , ) and improving previous lower bounds for other pairs.

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Local Properties via Color Energy Graphs and Forbidden Configurations

Cited by 12 publications

References 14 publications

A construction for difference sets with local properties

A construction for difference sets with local properties

The Erdős–Gyárfás function with respect to Gallai‐colorings

Lower bounds on the Erdős–Gyárfás problem via color energy graphs

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