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The size-Ramsey number of short subdivisions
Random Structures and Algorithms ( IF 0.9 ) Pub Date : 2021-01-11 , DOI: 10.1002/rsa.20995
Nemanja Draganić 1 , Michael Krivelevich 2 , Rajko Nenadov 1
Affiliation  

The r-size-Ramsey number urn:x-wiley:rsa:media:rsa20995:rsa20995-math-0001 of a graph H is the smallest number of edges a graph G can have such that for every edge-coloring of G with r colors there exists a monochromatic copy of H in G. For a graph H, we denote by Hq the graph obtained from H by subdividing its edges with q − 1 vertices each. In a recent paper of Kohayakawa, Retter and Rödl, it is shown that for all constant integers q, r ≥ 2 and every graph H on n vertices and of bounded maximum degree, the r-size-Ramsey number of Hq is at most urn:x-wiley:rsa:media:rsa20995:rsa20995-math-0002, for n large enough. We improve upon this result using a significantly shorter argument by showing that urn:x-wiley:rsa:media:rsa20995:rsa20995-math-0003 for any such graph H.

中文翻译:

短细分的大小-拉姆齐数

所述ř -size-Ramsey数urn:x-wiley:rsa:media:rsa20995:rsa20995-math-0001的曲线图的ħ是边缘的最小数量的图ģ可以有这样,对于每一个边染色的ģř颜色存在的单色副本ħģ。对于图H,我们用H q表示通过将其边细分为q  − 1 个顶点而从H获得的图。在 Kohayakawa、Retter 和 Rödl 最近的一篇论文中,表明对于所有常数整数q,  r  ≥ 2 和n上的每个图H顶点和有界最大度数,对于n足够大,H qr -size-Ramsey 数至多为。我们通过显示任何这样的图H使用明显更短的参数来改进这个结果。urn:x-wiley:rsa:media:rsa20995:rsa20995-math-0002urn:x-wiley:rsa:media:rsa20995:rsa20995-math-0003
更新日期:2021-01-11
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