Helly type problems in convexity spaces
Andreas F. Holmsen
TL;DR
This work investigates convexity spaces with bounded Radon number $r(X,\mathcal{C})$ and shows how this bound yields a network of equivalent Helly-type properties: colorful Helly $h_c$, fractional Helly $h_f$, and weak $\varepsilon$-nets, with $(p,q)$-theorems arising from these implications. It surveys the combinatorial foundations linking Radon, Helly, and Tverberg-type results, giving explicit bounds such as $h_c \le r^{r^{\log r}}$ and $t_k(X,\mathcal{C}) \le r^{r^{r^{\log r}}} \cdot k$, and discusses their sharpness and limitations. The second part introduces a novel construction: associated convexity spaces from uniform hypergraphs, and proves that the hypergraph class $\mathcal{H}_k(m)$ is $\chi$-bounded for all $m\ge k\ge 2$, via a chain of reductions involving colorful/fractional Helly numbers and Ramsey-type arguments, plus a direct bound on the Radon number. These results illuminate how convexity-space techniques can yield new hypergraph coloring bounds and suggest rich directions for extending Helly-type theory to broader combinatorial settings, including separable spaces and broader hypergraph families.
Abstract
We report on some recent progress regarding combinatorial properties in convexity spaces with a bounded Radon number. In particular, we discuss the relationship between the Radon number, the colorful and fractional Helly properties, weak $\varepsilon$-nets, and $(p,q)$-theorems. As an application of the theory of convexity spaces we introduce new classes of uniform hypergraphs and show that they are $χ$-bounded.