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Sub-Ramsey Numbers for Matchings

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Abstract

Given a graph G and a positive integer k, the sub-Ramsey number sr(G, k) is defined to be the minimum number m such that every \(K_{m}\) whose edges are colored using every color at most k times contains a subgraph isomorphic to G all of whose edges have distinct colors. In this paper, we will concentrate on \(sr(nK_{2},k)\) with \(nK_{2}\) denoting a matching of size n. We first give upper and lower bounds for \(sr(nK_{2},k)\) and exact values of \(sr(nK_{2},k)\) for some n and k. Afterwards, we show that \(sr(nK_{2},k)=2n\) when n is sufficiently large and \(k<\frac{n}{8}\) by applying the Local Lemma.

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Acknowledgements

Supported by NSFC (nos. 11671320, 11601429 and U1803263) and the Fundamental Research Funds for the Central Universities (no. 3102019GHJD003).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, China

    Fangfang Wu, Shenggui Zhang & Binlong Li

  2. Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, China

    Fangfang Wu, Shenggui Zhang & Binlong Li

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  1. Fangfang Wu
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  2. Shenggui Zhang
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Correspondence to Shenggui Zhang.

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Wu, F., Zhang, S. & Li, B. Sub-Ramsey Numbers for Matchings. Graphs and Combinatorics 36, 1675–1685 (2020). https://doi.org/10.1007/s00373-020-02216-2

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