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Szabo, Ross

doi:10.1037/e412132005-003

Cited by 2 publications

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“…For r = 2, they found a somewhat better construction to establish f 2 (n) n 0.4118 . This upper bound was subsequently improved to f 2 (n) n 0.3878 by Szabó [4,18]. Our first result shows that Proposition 1.1 is essentially sharp for r = 2, i.e.…”

Section: Introductionmentioning

confidence: 72%

Improved Ramsey-type results for comparability graphs

Korándi

Tomon

2020

Combinator. Probab. Comp.

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Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size $n^{1/(r+1)}$ . This bound is known to be tight for $r=1$ . The question whether it is optimal for $r\ge 2$ was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for $r=2$ , as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size $n^{1/3+o(1)}$ . Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G nor its complement contain a complete bipartite graph with parts of size $cn/{(log n)^r}$ . With this, we improve a result of Fox and Pach.

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Section: Introductionmentioning

confidence: 72%

Improved Ramsey-type results for comparability graphs

Korándi

Tomon

2020

Combinator. Probab. Comp.

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“…For r = 2, they found a somewhat better construction to establish f 2 (n) ≤ n 0.4118 . This upper bound was subsequently improved to f 2 (n) ≤ n 0.3878 by Szabó [17,4]. Our first result shows that Proposition 1 is essentially sharp for r = 2, i.e., f 2 (n) = n 1/3+o (1) .…”

Section: Introductionmentioning

confidence: 75%

Improved Ramsey-type results for comparability graphs

Korándi,

Tomon

2018

Preprint

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Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size n 1/(r+1) .This bound is known to be tight for r = 1. The question whether it is optimal for r ≥ 2 was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for r = 2, as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size n 1/3+o(1) .Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G, nor its complement contains a complete bipartite graph with parts of size cn (log n) r . With this, we improve a result of Fox and Pach.

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My Personal Story

Cited by 2 publications

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Improved Ramsey-type results for comparability graphs

Improved Ramsey-type results for comparability graphs

Improved Ramsey-type results for comparability graphs

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