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arXiv · 1601.01493

A note on induced Ramsey numbers

David Conlon·Domingos Dellamonica Jr.·Steven La Fleur·Vojtěch Rödl·Mathias Schacht
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Abstract

The induced Ramsey number $r_{\mathrm{ind}}(F)$ of a $k$-uniform hypergraph $F$ is the smallest natural number $n$ for which there exists a $k$-uniform hypergraph $G$ on $n$ vertices such that every two-coloring of the edges of $G$ contains an induced monochromatic copy of $F$. We study this function, showing that $r_{\mathrm{ind}}(F)$ is bounded above by a reasonable power of $r(F)$. In particular, our result implies that $r_{\mathrm{ind}}(F) \leq 2^{2^{ct}}$ for any $3$-uniform hypergraph $F$ with $t$ vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on an application of the hypergraph container method.

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David Conlon, Domingos Dellamonica Jr., Steven La Fleur, Vojtěch Rödl, Mathias Schacht. 2016-01-07. A note on induced Ramsey numbers. https://doi.org/10.1007/978-3-319-44479-6_13

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