Graphs with asymmetric Ramsey properties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915598696448000 |
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| author | Mendonça, Walner Miralaei, Meysam Mota, Guilherme O. |
| author_facet | Mendonça, Walner Miralaei, Meysam Mota, Guilherme O. |
| contents | Given positive integers $k$ and $\ell$ we write $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we denote by $R(k)$ the minimum $n$ such that $K_n\rightarrow (K_k,K_k)$. By using probabilistic methods and hypergraph containers we prove that for every integer $k \geq 3$, there exists a graph $G$ such that $G \nrightarrow (K_k,K_k)$ and $G \rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as a variation of a classical theorem of Nešetřil and Rödl [The Ramsey property for graphs with forbidden complete subgraphs, Journal of Combinatorial Theory, Series B, 20 (1976), 243-249], who proved that for every integer $k\geq 2$ there exists a graph $G$ with no copies of $K_k$ such that $G\rightarrow(K_{k-1}, K_{k-1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02963 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graphs with asymmetric Ramsey properties Mendonça, Walner Miralaei, Meysam Mota, Guilherme O. Combinatorics Given positive integers $k$ and $\ell$ we write $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we denote by $R(k)$ the minimum $n$ such that $K_n\rightarrow (K_k,K_k)$. By using probabilistic methods and hypergraph containers we prove that for every integer $k \geq 3$, there exists a graph $G$ such that $G \nrightarrow (K_k,K_k)$ and $G \rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as a variation of a classical theorem of Nešetřil and Rödl [The Ramsey property for graphs with forbidden complete subgraphs, Journal of Combinatorial Theory, Series B, 20 (1976), 243-249], who proved that for every integer $k\geq 2$ there exists a graph $G$ with no copies of $K_k$ such that $G\rightarrow(K_{k-1}, K_{k-1})$. |
| title | Graphs with asymmetric Ramsey properties |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.02963 |