Asymptotics of m(n)
Erdős (1963) proved that for any collection of fewer than sets of size n, there exists a 2-coloring in which no set is monochromatic. The proof is simple: Consider a random coloring. The probability that any one set is monochromatic is . By a union bound, the probability that any set is monochromatic is less than . Therefore, there exists a good coloring.
Erdős (1964) constructed an n-uniform graph with edges which does not have property B, establishing an upper bound. Schmidt (1963) proved that every collection of at most has property B. Erdős and Lovász conjectured that . Beck in 1978 improved the lower bound to . In 2000, Radhakrishnan and Srinivasan improved the lower bound to . They used a clever probabilistic algorithm.
Read more about this topic: Property B