The size‐Ramsey number of short subdivisions

نویسندگان

چکیده

The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest edges $G$ can have, such that for every edge-coloring with $r$ colors there exists monochromatic copy in $G$. notion size-Ramsey numbers has been introduced by Erdős, Faudree, Rousseau and Schelp 1978, attracted lot attention ever since. For $H$, we denote $H^q$ obtained from subdividing its $q{-}1$ vertices each. In recent paper Kohayakawa, Retter R{o}dl, it shown all constant integers $q,r\geq 2$ on $n$ bounded maximum degree, at most $(\log n)^{20(q-1)}n^{1+1/q}$, large enough. We improve upon this result using significantly shorter argument showing $\hat{R}_r(H^q)\leq O(n^{1+1/q})$ any $H$.

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ژورنال

عنوان ژورنال: Random Structures and Algorithms

سال: 2021

ISSN: ['1042-9832', '1098-2418']

DOI: https://doi.org/10.1002/rsa.20995