A superlinear lower bound for Radon Numbers

作者：Skand Parvatikar · 年份：2026 · 研究领域：Mathematics

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 P\'alv\"olgyi) that $r_k(X) = O_{r(X)}(k)$. P\'alv\"olgyi 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 $$ r(X_d)=\Theta(\log d) \qquad\text{and}\qquad r_d(X_d)=\Theta\bigl(d(\log d)^2\bigr) =\Theta\bigl(d\,r(X_d)^2\bigr). $$ The construction is a space similar to discrete box convexity, and the lower bound is obtained by random choice.