The Matching Ramsey Number of Hypergraphs, Revisited

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 $\tau (x)$, where the set of all such frequencies is bounded. We define a new parameter called the {\it $\tau$-matching chromatic number}, denoted by $\chi_M(\tau, {\mathcal H})$, 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 $\chi \left( {\rm KG}^r({\mathcal H}) \right)$ is a special case of $\chi_M(\tau, {\mathcal H})$ when all color frequencies are the fixed constant $r-1$. In this paper, we establish sharp lower bounds for the parameter $\chi_M(\tau , {\mathcal H})$, utilizing the concepts of the alternation number and the equitable colorability defect.