The Matching Ramsey Number of Hypergraphs, Revisited

Abstract

Suppose that a hypergraph $\mathcal{H}$ and an arbitrary nonempty (finite or infinite) set of available colors are given. Each color $x$ is associated with a frequency $\ddot{I}„(x)$, where the set of all such frequencies is bounded. We define a new parameter called the {\it $\ddot{I}„$-matching chromatic number}, denoted by $\ddot{I}{‡}_M(\ddot{I}„,\mathcal{H})$, as the least possible number of colors required to color the edges of $\mathcal{H}$ in such a way that the size of each nonempty monochromatic matching does not exceed the frequency of the corresponding color associated to its edges. The well-known and extensively well-studied chromatic number of general Kneser hypergraph $\ddot{I}‡\left({\mathrm{KG}}^r(\mathcal{H})\right)$ is a special case of $\ddot{I}{‡}_M(\ddot{I}„,\mathcal{H})$ when all color frequencies are the fixed constant $r-1$. In this paper, we establish sharp lower bounds for the parameter $\ddot{I}{‡}_M(\ddot{I}„,\mathcal{H})$, utilizing the concepts of the alternation number and the equitable colorability defect.