Recent Progress in Ramsey Theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913767944617984 |
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| author | Verstraete, Jacques |
| author_facet | Verstraete, Jacques |
| contents | The classical Ramsey numbers $r(s,t)$ denote the minimum $n$ such that every red-blue coloring of the edges of the complete graph $K_n$ contains either a red clique of order $s$ or a blue clique of order $t$. These quantities are the centerpiece of graph Ramsey Theory, and have been studied for almost a century. The Erdős-Szekeres Theorem (1935) shows that for each $s \geq 2$, $r(s,t) = O(t^{s - 1})$ as $t \rightarrow \infty$. We introduce a new approach using pseudorandom graphs which shows $r(4,t) = Ω(t^3/(\log t)^4)$ as $t \rightarrow \infty$, answering an old conjecture of Erdős, and we illustrate how to apply this approach to many other Ramsey and related combinatorial problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22094 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recent Progress in Ramsey Theory Verstraete, Jacques Combinatorics 05C The classical Ramsey numbers $r(s,t)$ denote the minimum $n$ such that every red-blue coloring of the edges of the complete graph $K_n$ contains either a red clique of order $s$ or a blue clique of order $t$. These quantities are the centerpiece of graph Ramsey Theory, and have been studied for almost a century. The Erdős-Szekeres Theorem (1935) shows that for each $s \geq 2$, $r(s,t) = O(t^{s - 1})$ as $t \rightarrow \infty$. We introduce a new approach using pseudorandom graphs which shows $r(4,t) = Ω(t^3/(\log t)^4)$ as $t \rightarrow \infty$, answering an old conjecture of Erdős, and we illustrate how to apply this approach to many other Ramsey and related combinatorial problems. |
| title | Recent Progress in Ramsey Theory |
| topic | Combinatorics 05C |
| url | https://arxiv.org/abs/2503.22094 |