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Abstract

In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals. Helly, Carathéodory, and Radon theoremsIn this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg.Helly's theorem [Hel23] asserts that for a family {K 1 , K 2 , . . . , K n } of convex sets in R d , where n ≥ d + 1, if every d + 1 of the sets have a point in common, then all of the sets have a point in common. The closely related Carathéodory theorem [Car07] states that forThe more general colorful Carathéodory theorem [Bár82] says the following. Let S 1 , S 2 , . . . , S d+1 be d + 1 sets (or colors if you wish) in R d . Suppose that x ∈ d+1 i=1 conv S i . Then there is a transversal T = {x 1 , . . . , x d+1 } of the system S 1 , . . . , S d+1 , meaning that x 1 ∈ S 1 , x 2 ∈ S 2 , . . . , x d+1 ∈ S d+1 such that x ∈ conv T . A transversal is also called a rainbow set when S 1 , . . . , S d+1 are considered as colors. The uncolored version, that is, when S 1 = S 2 = • • • = S d+1 , is the classic result of Carathéodory. There is a closely related colorful version of Helly's theorem due to Lovász that appeared in [Bár82].Tverberg's theorem [Tve66] states the following. Let x 1 , x 2 , . . . , x m be points in R d with m ≥ (r − 1)(d + 1) + 1. Then there is a partition S 1 , S 2 , . . . , S r of {1, 2, . . . , m} such that r j=1 conv {x i : i ∈ S j } = ∅. This was a conjecture by Birch, who also proved the planar case in a slightly different form. The bound of (r −1)(d+1)+1 in the theorem is sharp as can easily be seen from the configuration of points in a sufficiently general position.The case r = 2 is Radon's theorem [Rad21], another classic from 1921, which was used by Radon to prove Helly's theorem. Helly's original proof (published later) was based on a separation argument. Sarkaria [Sar92] gave a simple proof of Tverberg's theorem based on the colorful Carathéodory theorem.