Capturing a phylogenetic tree when the number of character states varies with the number of leaves

We show that for any two values $α, β>0$ for which $α+β>1$ then there is a value $N$ so that for all $n \geq N$ the following holds. For any binary phylogenetic tree $T$ on $n$ leaves there is a set of $\lfloor n^α\rfloor$ characters that capture $T$, and for which each character takes at most $\lfloor n^β\rfloor$ distinct states. Here `capture' means that $T$ is the unique perfect phylogeny for these characters. Our short proof of this combinatorial result is based on the probabilistic method.

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