Counterexamples to the Corsten-Frankl conjecture on diameter-Ramsey simplices

Abstract

Corsten and Frankl conjectured that a simplex is diameter-Ramsey if and only if its circumcenter lies in its convex hull. We disprove this conjecture in every dimension $d\ge 3$. The main tool is a sufficient criterion based on a higher-order deficit decomposition: if the squared deficits $D^2-\Vert p_i-p_j{\Vert }^2$ admit a nonnegative decomposition over subsets of the vertex set, with total mass at most $D^2$, then the simplex is diameter-Ramsey. The pairwise deficit criterion of Frankl--Pach--Reiher--Rödl is recovered as a special case. As an application, for every $d\ge 3$ we construct a diameter-Ramsey $d$-simplex whose circumcenter lies outside its convex hull. A particularly simple family has squared edge lengths $\Vert p_1-p_2{\Vert }^2=\Vert p_1-p_j{\Vert }^2=7(4\le j\le d+1)$, $\Vert p_1-p_3{\Vert }^2=4$, and $\Vert p_i-p_j{\Vert }^2=4(2\le i<j\le d+1)$.

11 pages