2018
DOI: 10.48550/arxiv.1810.00588
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Improved Ramsey-type results for comparability graphs

Dániel Korándi,
István Tomon

Abstract: 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. … Show more

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Cited by 2 publications
(2 citation statements)
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“…However, this alone will not be sufficient to find a linear-sized homogeneous submatrix. Indeed, as was shown recently by Korándi and Tomon[21], the size of the bipartite graph in Theorem 6.1 cannot be replaced by anything larger than Ω(n/(log n) k ).However, one can find slightly larger all-0 submatrices in Lemma 6.2 by reducing the number of partial orders we use. For example, we may assume that K in the proof has size at most k − 1, as…”
mentioning
confidence: 77%
“…However, this alone will not be sufficient to find a linear-sized homogeneous submatrix. Indeed, as was shown recently by Korándi and Tomon[21], the size of the bipartite graph in Theorem 6.1 cannot be replaced by anything larger than Ω(n/(log n) k ).However, one can find slightly larger all-0 submatrices in Lemma 6.2 by reducing the number of partial orders we use. For example, we may assume that K in the proof has size at most k − 1, as…”
mentioning
confidence: 77%
“…The rest of this section is devoted to the proof of this theorem. The proof is probabilistic and is inspired by a construction of Korándi and Tomon [20]. We shall consider a random double-ordered…”
Section: It Easily Follows From the Above Definition That Ifmentioning
confidence: 99%