Journal of Combinatorial Theory Series B ( IF 1.4 ) Pub Date : 2020-09-16 , DOI: 10.1016/j.jctb.2020.09.004 Louis DeBiasio , András Gyárfás , Gábor N. Sárközy
A path-matching of order p is a vertex disjoint union of nontrivial paths spanning p vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers , define to be the smallest integer n such that in any r-coloring of the edges of there exists a path-matching of color i and order at least for some . Our main result is that for and , if , then Perhaps surprisingly, we show that when , it is possible that is larger than , but in any case we determine the correct value to within a constant (depending on r); i.e. As a corollary we get that in every r-coloring of there is a monochromatic path-matching of order at least , which is essentially best possible. We also determine in all cases when the number of colors is at most 4.
The proof of the main result uses a minimax theorem for path-matchings derived from a result of Las Vergnas (extending Tutte's 1-factor theorem) to show that the value of depends on the block sizes in covering designs (which can be also formulated in terms of monochromatic 1-cores in colored complete graphs). While block sizes in covering designs have been studied intensively before, they seem to have only been studied in the uniform case (when all block sizes are equal). Then we obtain the result above by giving estimates on the block sizes in covering designs in the arbitrary (non-uniform) case.
中文翻译:
路径匹配,覆盖设计和1核的Ramsey数量
阶数为p的路径匹配是跨越p个顶点的非平凡路径的顶点不相交的并集。Burr和Roberts,以及Faudree和Schelp确定了路径匹配的2色Ramsey数。在本文中,我们研究了路径匹配的多色Ramsey数。给定正整数,定义 是最小的整数n,使得在任何r的边缘着色存在颜色i的路径匹配并且至少排序 对于一些 。我们的主要结果是 和 如果 , 然后 也许令人惊讶的是,我们表明 , 它可能是 大于 ,但无论如何,我们都会在常数内确定正确的值(取决于r);即作为一个推论,我们得到的是在每一个[R -coloring的 至少存在阶的单色路径匹配 ,这实际上是最好的选择。我们还确定 在所有情况下,颜色数量最多为4。
主要结果的证明对Las Vergnas(扩展了Tutte的1因子定理)的结果得出的路径匹配使用极小极大定理,以证明 取决于覆盖设计中的块大小(也可以用彩色完整图中的单色1芯表示)。尽管之前已经对覆盖设计中的块大小进行了深入研究,但似乎仅在统一情况下(所有块大小相等时)进行了研究。然后,通过对任意(非均匀)情况下的覆盖设计中的块大小进行估计,可以得出上述结果。