Improved Lower Bounds for Property B

If an n-uniform hypergraph can be 2-colored, then it is said to have property B. Erd{o}s (1963) was the first to give lower and upper bounds for the minimal size $m(n)$ of an n-uniform hypergraph without property B. His asymptotic upper bound O(n²2ⁿ) still is the best we know, his lower bound 2ⁿ⁻¹ has seen a number of improvements, with the current best Ω (2ⁿ√n/log(n)) established by Radhakrishnan and Srinivasan (2000). Cherkashin and Kozik (2014) provided a simplified proof of this result, using Pluh\'ar's (2009) idea of a random greedy coloring. In the present paper, we use a refined version of this argument to obtain improved lower bounds on $m(n)$ for small values of n. We also study $m(n,v)$, the size of the smallest n-hypergraph without property B having v vertices.