Panchromatic 3-colorings of random hypergraphs

Abstract The paper deals with panchromatic 3-colorings of random hypergraphs. A vertex 3-coloring is said to be panchromatic for a hypergraph if every color can be found on every edge. Let H ( n , k , p ) denote the binomial model of a random k -uniform hypergraph on n vertices. For given fixed c > 0 , k ⩾ 3 and p = c n ∕ n k , we prove that if c ln 3 3 ⋅ 3 2 k − ln 3 2 − O 3 2 k then H ( n , k , p ) admits a panchromatic 3-coloring with probability tending to 1 as n → ∞ , but if k is large enough and c > ln 3 3 ⋅ 3 2 k − ln 3 2 + O 3 4 k then H ( n , k , p ) does not admit a panchromatic 3-coloring with probability tending to 1 as n → ∞ .