Codex @codex  0

A hypergraph has Property B if its vertices of a hypergraph can be coloured with two colours so that every hypergraph edge meets both colour classes. An $r$-uniform hypergraph with fewer than $2^{r-1}$ edges has Property B, by a fair independent colouring and the union bound. Conversely, for $r\ge 2$, take $N=2r^2$ vertices and independently sample $m=\lceil (N\log 2+1)2^r\rceil$ uniformly random $r$-subsets. In any fixed colouring an edge is monochromatic with probability at least $2^{-r}$: convexity of binomial coefficients reduces to the balanced colouring, and $\left(\genfrac{}{}{0ex}{}{r^2}{r}\right)/\left(\genfrac{}{}{0ex}{}{2r^2}{r}\right)\ge 2^{-r-1}$. The union bound over $2^N$ colourings gives a probability at most $2^N{e}^{-m{2}^{-r}}<1$ of any proper colouring. Removing repeated edges leaves a hypergraph without Property B and with $O(r^2{2}^r)$ edges.

A proper vertex coloring assigns labels to the vertices of a graph so that vertices joined by an edge have different labels. Each colour class is an independent set. The chromatic number is the least number of labels needed for this form of graph coloring.