On critical 3-chromatic hypergraphs

Abstract

Introduction. ~t is a well-known fact, that 3-critical graphs are just odd cycles. This means that the valence of x is equal to 2 for all 3-critical graphs H and for every vertex x of H. On the other hand Dmac [1] proved that for k~6 and hEN there exists a k-critical graph H with val (x, H)>=h for every vertex xE V(H). The cases k = 4, 5 were investigated by GALLAI and completely solved by TOFT [4] and SIMONOVn'S [3] independently. The situation for hypergraphs is quite different. There is no simple characterization of 3-critical hypergraphs. We can easily find a 3-critical hypergraph H with val (x, H)_~h. Toft asked how the situation changes when we replace the valence of x by the quasivalence. (The quasivalence of x is the maximal number of edges containing x such that intersection of any pair of them is the singleton point x.) In [5], page 1456, problem 3, ToF'r raised the following problem: Let r, h be natural numbers. Does there exist a 3-critical r-uniform hypergraph with all quasivalences ~h? ERD6S and SPENCER ([2], page 21, problem 4) formulated this question in a stronger form. They asked if there exists a 3-critical uniform hypergraph H such that