Induced subgraph density. V. All paths approach Erdos-Hajnal
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929550555873280 |
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| author | Nguyen, Tung Scott, Alex Seymour, Paul |
| author_facet | Nguyen, Tung Scott, Alex Seymour, Paul |
| contents | The Erdős-Hajnal conjecture says that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a clique or stable set of size at least $n^c$. In this paper we are concerned with the case when $H$ is a path. The conjecture has been proved for paths with at most five vertices, but not for longer paths. We prove that the conjecture is ``nearly'' true for all paths: for every path $H$, all $H$-free graphs with $n$ vertices have cliques or stable sets of size at least $2^{(\log n)^{1-o(1)}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15032 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Induced subgraph density. V. All paths approach Erdos-Hajnal Nguyen, Tung Scott, Alex Seymour, Paul Combinatorics The Erdős-Hajnal conjecture says that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a clique or stable set of size at least $n^c$. In this paper we are concerned with the case when $H$ is a path. The conjecture has been proved for paths with at most five vertices, but not for longer paths. We prove that the conjecture is ``nearly'' true for all paths: for every path $H$, all $H$-free graphs with $n$ vertices have cliques or stable sets of size at least $2^{(\log n)^{1-o(1)}}$. |
| title | Induced subgraph density. V. All paths approach Erdos-Hajnal |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.15032 |