OAI.RyserOdd.eventualOddFailures_and_infinite

The theorem states that two things hold. First, there is a threshold N such that for every prime p ≥ N with p ≡ 2 (mod 3), there is a degree threshold D ≥ 3 such that for every odd d ≥ D, ExplicitFailureAt(p^d + 1) holds. Second, the set of ranks r for which ExplicitFailureAt(r) holds is infinite. Here ExplicitFailureAt(r) means that r > 0 and there exist a finite type of vertices with decidable equality and a nonempty hypergraph (a finset of finsets of vertices) on it that is r-partite (some map from vertices to Fin r makes each edge contain exactly one vertex from every part), r-uniform (every edge has exactly r vertices), and intersecting (any two distinct edges share a vertex), and that has at least one vertex cover (a vertex set meeting every edge). Moreover its matching number, the largest size of a family of pairwise disjoint edges, equals 1, and its cover number, the least size of a vertex cover, equals r. Consequently (r − 1)·(matching number) < cover number, so the hypergraph violates the Ryser-type bound cover number ≤ (r − 1)·matching number.