Counterexamples to the Corsten-Frankl conjecture on diameter-Ramsey simplices
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908983013408768 |
|---|---|
| author | Mao, Yaping |
| author_facet | Mao, Yaping |
| contents | 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 $D^2-\|p_i-p_j\|^2$ 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 $\|p_1-p_2\|^2=\|p_1-p_j\|^2=7~ (4\le j\le d+1)$, $\|p_1-p_3\|^2=4$, and $\|p_i-p_j\|^2=4 ~ (2\le i<j\le d+1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19126 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counterexamples to the Corsten-Frankl conjecture on diameter-Ramsey simplices Mao, Yaping Combinatorics 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 $D^2-\|p_i-p_j\|^2$ 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 $\|p_1-p_2\|^2=\|p_1-p_j\|^2=7~ (4\le j\le d+1)$, $\|p_1-p_3\|^2=4$, and $\|p_i-p_j\|^2=4 ~ (2\le i<j\le d+1)$. |
| title | Counterexamples to the Corsten-Frankl conjecture on diameter-Ramsey simplices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.19126 |