An Explicit $O(r\log r)$ Threshold for Attaining the Semple--Steel Bound with $r$-State Characters

Abstract

Let $d_r(n)$ be the maximum, over all binary phylogenetic trees with $n$ leaves, of the minimum number of $r$-state characters required to define the tree. Semple and Steel proved that $d_r(n)\ge \lceil (n-3)/(r-1)\rceil$, and Bordewich and Semple proved that equality holds for each fixed $r$ and all sufficiently large $n$. We study the corresponding threshold $n_r$, the least $N$ for which equality holds for every $n\ge N$. The Bordewich--Semple construction yields an explicit polynomial upper bound of order $O(r^5)$ for this threshold. We prove the near-linear estimate \[ 3r+1\leq n_r\leq \ceil{64(r-1)\log_2(r+1)}+3\qquad(r\geq4). \] The proof constructs, for every binary phylogenetic tree with $m=n-3$ internal edges, a linked quartet certificate whose conflict graph has maximum degree at most $16\lceil {\log }_2(m+2)\rceil +4$. Equitable coloring then packs the certificate into exactly $\lceil m/(r-1)\rceil$ $r$-state characters once $m\ge \lceil 64(r-1){\log }_2(r+1)\rceil$. We also include the lower bound $n_r\ge 3r+1$, obtained from the snowflake obstruction, and state the natural conjecture that this lower bound is the exact threshold for all $r\ge 4$. The conjectural endpoint is consistent with the known small-state thresholds: $n_4=13$ and $n_5=16$, while the cases $r=2,3$ are also explicitly classified.