A superlinear lower bound for Radon Numbers

Skand Parvatikar

paper

A superlinear lower bound for Radon Numbers

arXiv:2608.06437

Abstract

For a convexity space $X$ , let $r (X) = r_{2} (X)$ be its Radon number and let $r_{k} (X)$ be its $k$ -th Radon number. It is known (due to Pálvölgyi) that $r_{k} (X) = O_{r (X)} (k)$ . Pálvölgyi asked if there was an absolute constant $C$ such that for every abstract convexity space $X$ , $r_{k} (X) \leq C r (X) k$ . We resolve this question in the negative. More precisely, we construct finite convexity spaces $X_{d}$ satisfying $r (X_{d}) = Θ (lo g d) and r_{d} (X_{d}) = Θ (d (lo g d)^{2}) = Θ (d r (X_{d})^{2}) .$ The construction is a space similar to discrete box convexity, and the lower bound is obtained by random choice.

3 pages, comments welcome!