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- W4300625557 abstract "Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $rge 2$. We define the ($t$-color) chromatic Ramsey number $chi(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph are colored with $t$ colors in any manner, there is a monochromatic copy of $T$. We observe that $chi(T,t)$ is well defined and $$leftlceil {R^r(T,t)-1over r-1}right rceil +1 le chi(T,t)le |E(T)|^t+1$$ where $R^r(T,t)$ is the $t$-color Ramsey number of $H$. We give linear upper bounds for $chi(T,t)$ when T is a matching or star, proving that for $rge 2, kge 1, tge 1$, $chi(M_k^r,t)le (t-1)(k-1)+2k$ and $chi(S_k^r,t)le t(k-1)+2$ where $M_k^r$ and $S_k^r$ are, respectively, the $r$-uniform matching and star with $k$ edges. The general bounds are improved for $3$-uniform hypergraphs. We prove that $chi(M_k^3,2)=2k$, extending a special case of Alon-Frankl-Lov'asz' theorem. We also prove that $chi(S_2^3,t)le t+1$, which is sharp for $t=2,3$. This is a corollary of a more general result. We define $H^{[1]}$ as the 1-intersection graph of $H$, whose vertices represent hyperedges and whose edges represent intersections of hyperedges in exactly one vertex. We prove that $chi(H)le chi(H^{[1]})$ for any $3$-uniform hypergraph $H$ (assuming $chi(H^{[1]})ge 2$). The proof uses the list coloring version of Brooks' theorem." @default.
- W4300625557 created "2022-10-03" @default.
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- W4300625557 date "2015-09-01" @default.
- W4300625557 modified "2023-10-16" @default.
- W4300625557 title "Chromatic Ramsey number of acyclic hypergraphs" @default.
- W4300625557 doi "https://doi.org/10.48550/arxiv.1509.00551" @default.
- W4300625557 hasPublicationYear "2015" @default.
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