Abstract
Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.
Highlights
Given a finite point set in Rd the intersection pattern of convex hulls determined by subsets of those points is the focus of Tverberg-type theory
Many seemingly simple questions of Tverberg-type remain open — among them a conjecture of Reay [17]: for any r ≥ 2 and d ≥ 1 there are (r − 1)(d + 1) points in Rd such that for any partition of them into r parts, two of them have disjoint convex hulls. This would imply that there is no relaxation of Tverberg’s theorem, where fewer than (r − 1)(d + 1) + 1 points can be partitioned into r sets of pairwise intersecting convex hulls. This problem has been studied for k-fold intersections among the r convex hulls instead of only pairwise intersections
If we are given that all facets have nonempty pairwise intersections, how does this restrict the possible combinatorics of K? In the special case of graphs this would be answered by Conway’s thrackle conjecture: a thrackle is a graph that can be drawn in the plane in such a way that any pair of edges intersects precisely once, either at a common vertex or a transverse intersection point
Summary
Given a finite point set in Rd the intersection pattern of convex hulls determined by subsets of those points is the focus of Tverberg-type theory. Many seemingly simple questions of Tverberg-type remain open — among them a conjecture of Reay [17]: for any r ≥ 2 and d ≥ 1 there are (r − 1)(d + 1) points in Rd such that for any partition of them into r parts, two of them have disjoint convex hulls This would imply that there is no relaxation of Tverberg’s theorem, where fewer than (r − 1)(d + 1) + 1 points can be partitioned into r sets of pairwise intersecting convex hulls. We isolate a feature of the pairwise intersection pattern of convex sets that allows us to prove an extension of the linear thrackle conjecture: we establish the bound m ≤ n if the full-dimensional Ci are vertex-disjoint from one another and there is a transversal set W that contains all vertices and possibly more points such that |Ci ∩ Cj ∩ W | = 1 for all i = j; see Theorem 4.3. We present higher-dimensional generalizations of the linear thrackle conjecture in Section 5 and conjecture their continuous analogs
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