On a Ramsey-type problem of Erd\H{o}s and Pach

Ross J. Kang; Eoin Long; Viresh Patel; Guus Regts

doi:10.1112/blms.12094

On a Ramsey-type problem of Erd\H{o}s and Pach

Combinatorics 2017-10-18 v2

Authors: Ross J. Kang , Eoin Long , Viresh Patel , Guus Regts

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Abstract

In this paper we show that there exists a constant $C>0$ such that for any graph $G$ on $Ck\ln k$ vertices either $G$ or its complement $\bar{G}$ has an induced subgraph on $k$ vertices with minimum degree at least $\frac12(k-1)$ . This affirmatively answers a question of Erd\H{o}s and Pach from 1983.

Keywords

Ramsey-type problem induced subgraph minimum degree graph complement Erd\H{o}s and Pach question extremal graph theory

Cite

@article{arxiv.1411.4459,
  title  = {On a Ramsey-type problem of Erd\H{o}s and Pach},
  author = {Ross J. Kang and Eoin Long and Viresh Patel and Guus Regts},
  journal= {arXiv preprint arXiv:1411.4459},
  year   = {2017}
}

Comments

9 pages; in 2nd version, Eoin Long has been added as co-author and the main result is improved by a log k factor

On a Ramsey-type problem of Erd\H{o}s and Pach

Abstract

Keywords

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