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

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 $D^2-\mathrm{\parallel }{p}_i-p_j{\mathrm{\parallel }}^2$ 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 $\mathrm{\parallel }{p}_1-p_2{\mathrm{\parallel }}^2=\mathrm{\parallel }{p}_1-p_j{\mathrm{\parallel }}^2=7(4\le j\le d+1)$, $\mathrm{\parallel }{p}_1-p_3{\mathrm{\parallel }}^2=4$, and $\mathrm{\parallel }{p}_i-p_j{\mathrm{\parallel }}^2=4(2\le i<j\le d+1)$.