A note on random greedy coloring of uniform hypergraphs

The smallest number of edges forming an n-uniform hypergraph which is not r-colorable is denoted by m(n, r). Erdős and Lovász conjectured thatm(n, 2) = θ (n2n). The best known lower boundm(n, 2) = Ω (√ n/ logn2n ) was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on analysis of random greedy coloring algorithm investigated by Pluhár in 2009. The proof method extends to the case of r-coloring, and we show that for any fixed r we have m(n, r) = Ω ( (n/ logn)(r−1)/r rn ) improving the bound of Kostochka from 2004. We also derive analogous bounds on minimum edge degree of an nuniform hypergraph that is not r-colorable. A hypergraph is a pair (V,E), where V is a set of vertices and E is a family of subsets of V . Hypergraph is n-uniform if all its edges have exactly n elements. Hypergraph (V,E) is r colorable if there exists a coloring of vertices with r colors in which no edge is monochromatic (i.e., there exists a function c : V →{1, . . . , r} such that the image of every edge has at least two elements). Hypergraph has property B if it is two-colorable. For n, r ∈ N let m(n, r) be the smallest number of edges of an n-uniform hypergraph that is not r-colorable. The asymptotic behaviour of m(n) = m(n, 2) was first studied by Erdős. In [1] and [2] Erdős proved that: 2n−1 6 m(n) 6 (1 + o(1)) e ln 2 4 n2. In [3] Erdős and Lovász wrote that “perhaps n2 is the correct order of magnitude of m(n)”. The upper bound has not been improved since. The most recent improvement on the lower bound was obtained by Radhakrishnan and Srinivasan in [7]. We present a simple proof of their main theorem: Theorem 1 (Radhakrishnan, Srinivasan 2000). m(n) = Ω (( n ln(n) )1/2 2 )