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arXiv · 1801.04128

Improved bounds on the multicolor Ramsey numbers of paths and even cycles

Charlotte Knierim·Pascal Su
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Abstract

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest.

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Charlotte Knierim, Pascal Su. 2018-01-12. Improved bounds on the multicolor Ramsey numbers of paths and even cycles. https://arxiv.org/abs/1801.04128

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