On the construction of non-22-colorable uniform hypergraphs

The problem of 22-coloring uniform hypergraphs has been extensively studied over the last few decades. An nn-uniform hypergraph is not 22-colorable if its vertices cannot be colored with two colors, Red and Blue, such that every hyperedge contains Red as well as Blue vertices. The least possible number of hyperedges in an nn-uniform hypergraph which is not 22-colorable is denoted by m(n)m(n). In this paper, we consider the problem of finding an upper bound on m(n)m(n) for small values of nn. We provide constructions which improve the existing results for some such values of nn. We obtain the first improvement in the case of n=8n=8.