Ramsey-type results for semi-algebraic relations

Conlon, David; Fox, Jacob; Pach, János; Sudakov, Benny; Suk, Andrew

doi:10.1145/2462356.2462399

Cited by 23 publications

(36 citation statements)

References 24 publications

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“…(We defined semilinearity only for graphs, but the definition extends for hypergraphs in a straightforward way.) It was proved by Conlon et al [9] that if H is an r-uniform semialgebraic hypergraph on N vertices of complexity t which contains no clique or independent set of size n, then N ≤ tw r−1 (n C ) for some constant C = C(t), where the tower function tw k (x) is defined as tw 1 (x) := x and tw k+1 (x) := 2 tw k (x) . This bound is also the best possible up to the value of C. Do the Ramsey numbers of r-uniform semilinear hypergraphs behave similarly?…”

Section: Discussionmentioning

confidence: 81%

Ramsey properties of semilinear graphs

Tomon¹

2021

Preprint

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A graph G is semilinear of complexity t if the vertices of G are elements of R d for some d ∈ Z + , and the edges of G are defined by the sign patterns of t linear functions f 1 , . . . , f t : R d × R d → R. We show that semilinear graphs of constant complexity have very tame Ramsey properties. More precisely, we prove that if G is a semilinear graph of complexity t which contains no clique of size s and no independent set of size n, then G has at most O s,t (n) • (log n) Ot(1) vertices. We also show that the logarithmic term cannot be omitted. In particular, this implies that if G is a semilinear graph of constant complexity on n vertices, and G contains no clique of size s, then G can be properly colored with polylog(n) colors. In the past 60 years, this coloring question was extensively studied for several special instances of semilinear graphs, e.g. shift graphs, intersection and disjointness graphs of certain geometric objects, and overlap graphs. Our main result provides a general upper bound on the chromatic number of all such, seemingly unrelated, graphs.Furthermore, we consider the symmetric Ramsey problem for semilinear graphs as well. It is known that if there exists an intersection graph of N boxes in R d (such graphs are semilinear of complexity 2d) that contains no clique or independent set of size n, then N = O d (n 2 (log n) d−1 ). That is, the exponent of n does not grow with the dimension. We prove a result about the symmetric Ramsey properties of semilinear graphs, which puts this phenomenon in a more general context.

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Section: Discussionmentioning

confidence: 81%

Ramsey properties of semilinear graphs

Tomon¹

2021

Preprint

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“…Over the past decade, several authors have shown that many classical theorems in extremal graph theory can be significantly improved if we restrict our attention to semi-algebraic graphs, that is, graphs whose vertices are points in Euclidean space, and edges are defined by a semi-algebraic relation of constant complexity [1,5,8,11,9,4]. In this note, we continue this sequence of works by studying Ramsey-Turán numbers for semi-algebraic graphs.…”

Section: Introductionmentioning

confidence: 88%

Ramsey-Turán Numbers for Semi-Algebraic Graphs

Fox¹,

Pach²,

Suk³

2018

Electron. J. Combin.

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A semi-algebraic graph $G = (V,E)$ is a graph where the vertices are points in $\mathbb{R}^d$, and the edge set $E$ is defined by a semi-algebraic relation of constant complexity on $V$. In this note, we establish the following Ramsey-Turán theorem: for every integer $p\geq 3$, every $K_{p}$-free semi-algebraic graph on $n$ vertices with independence number $o(n)$ has at most $\frac{1}{2}\left(1 - \frac{1}{\lceil p/2\rceil-1} + o(1) \right)n^2$ edges. Here, the dependence on the complexity of the semi-algebraic relation is hidden in the $o(1)$ term. Moreover, we show that this bound is tight.

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“…r (t) denote the smallest N such that any r-uniform semi-algebraic hypergraph of complexity (n, d, m) contains either a clique or an independent set of size t. Conlon et al [13] studied the Ramsey problem for semi-algebraic hypergraphs and proved that there exists c = c(r, n, m, d) > 0 and…”

Section: Ramsey Properties Of Algebraic Hypergraphsmentioning

confidence: 99%

Ramsey properties of algebraic graphs and hypergraphs

Sudakov¹,

Tomon²

2021

Preprint

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One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on N vertices. By the celebrated result of Erdős from 1947, the random graph on N vertices with edge probability 1/2, contains no clique or independent set larger than 2 log 2 N , with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools.Say that an r-uniform hypergraph H is algebraic of complexity (n, d, m) if the vertices of H are elements of F n for some field F, and there exist m polynomials f 1 , . . . , f m : (F n ) r → F of degree at most d such that the edges of H are determined by the zero-patterns of f 1 , . . . , f m . The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity (n, d, m) has good Ramsey properties, then at least one of the parameters n, d, m must be large.In 2001, Rónyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if G is an algebraic graph of complexity (n, d, m) on N vertices, then either G or its complement contains a complete balanced bipartite graph of size Ω n,d,m (N 1/(n+1) ). We extend this result by showing that such G contains either a clique or an independent set of size N Ω(1/ndm) and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for r-uniform algebraic hypergraphs that are defined by a single polynomial, that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.

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Ramsey-type results for semi-algebraic relations

Cited by 23 publications

References 24 publications

Ramsey properties of semilinear graphs

Ramsey properties of semilinear graphs

Ramsey-Turán Numbers for Semi-Algebraic Graphs

Ramsey properties of algebraic graphs and hypergraphs

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