Improved Lower Bounds for Property B

If an $n$-uniform hypergraph can be 2-colored, then it is said to have property B. Erdős (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 $O(n^2{2}^n)$ still is the best we know, his lower bound $2^{n-1}$ has seen a number of improvements, with the current best $\mathrm{\Omega }$ $(2^n\sqrt{n\mathrm{/}\log (n)})$ established by Radhakrishnan and Srinivasan (2000). Cherkashin and Kozik (2014) provided a simplified proof of this result, using Pluhár’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.