Bounding low-degree Betti numbers of all intersections of a set family bounds its Radon number, giving an optimal surface fractional Helly theorem for b=0.
Radon numbers and the fractional Helly theorem
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abstract
A basic measure of the combinatorial complexity of a convexity space is its Radon number. In this paper we show a fractional Helly theorem for convexity spaces with a bounded Radon number, answering a question of Kalai. As a consequence we also get a weak epsilon-net theorem for convexity spaces with a bounded Radon number. This answers a question of Bukh and extends a recent result of Moran and Yehudayoff.
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Bounding Radon numbers via Betti numbers
Bounding low-degree Betti numbers of all intersections of a set family bounds its Radon number, giving an optimal surface fractional Helly theorem for b=0.