BIRS Workshop Lecture Videos
Bounding Radon's number via Betti numbers in the plane. Patak, Pavel
In her recent result, Zuzana PatÃ¡kovÃ¡ has shown that for a finite family $\mathcal F$ of sets in $\mathbb R^d$, one can use Betti numbers of intersections of subfamilies of $\mathcal F$, to bound the Radon's number of $\mathcal F$. The result has interesting consequences, some of them are easy or standard, other follow from a result of Holmsen and Lee. Let me name just few: variants of Helly's, Tverberg's, colorful and fractional Helly theorems, existence of weak $\varepsilon$-nets, $(p,q)$-theorems, \dots Nevertheless, the original bounds on Radon's number are too large to be widely applicable. We improve the situation in the plane and show how to obtain polynomial bounds. More generally the result extends to other two-dimensional (pseudo)manifolds.
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