Some new bounds on partition critical hypergraphs

A hypergraph ( [ n ] , E ) is 3-color critical if it is not 2-colorable, but for all E ∈ E the hypergraph ( [ n ] , E ∖ { E } ) is 2-colorable. Lovász proved in 1976, that | E | ≤ n k − 1 if E is k -uniform. Here we give a new algebraic proof and an ordered version that is a sharpening of Lovász’ result. ⊆ [ n ] k be a k -uniform set system on an underlying set [ n ] of n elements. Let us fix an ordering E 1 , E 2 , … E t of E and a prescribed partition { A i , B i } of each E i (i.e., A i ∪ B i = E i and A i ∩ B i = 0̸ ). Assume that for all i = 1 , 2 , … , t there exists a partition { C i , D i } of [ n ] such that E i ∩ C i = A i and E i ∩ D i = B i , but { E j ∩ C i , E j ∩ D i } ≠ { A j , B j } for all j < i . That is, the i th partition cuts the i th set as it was prescribed, but it does not cut any earlier set properly. Then t ≤ f ( n , k ) : = n − 1 k − 1 + n − 1 k − 2 + ⋯ + n − 1 0 . This is sharp for k = 2 , 3 . We show that this upper bound is almost the best possible, at least the first three terms are correct; we give constructions of size f ( n , k ) − O ( n k − 4 ) (for k fixed and n → ∞ ). We also give constructions of sizes n k − 1 for all n and k . rmore, in the 3-color-critical case (i.e. { A i , B i } = { E i , 0̸ } for all i ), t ≤ n k − 1 .