Two Disjoint 5-Holes in Point Sets

Abstract

Given a set of points S ⊆ R 2 , a subset X ⊆ S, |X| = k, is called k-gon if all points of X lie on the boundary of the convex hull conv(X), and k-hole if, in addition, no point of S \ X lies in conv(X). We use computer assistance to show that every set of 17 points in general position admits two disjoint 5-holes, that is, holes with disjoint respective convex hulls. This answers a question of Hosono and Urabe (2001). We also provide new bounds for three and more pairwise disjoint holes.In a recent article, Hosono and Urabe ( 2018) present new results on interior-disjoint holes -a variant, which also has been investigated in the last two decades. Using our program, we show that every set of 15 points contains two interior-disjoint 5-holes.Moreover, our program can be used to verify that every set of 17 points contains a 6-gon within significantly smaller computation time than the original program by Szekeres and Peters (2006).1 Erdős offered $500 for a proof of Szekeres' conjecture that g(k) = 2 k−2 + 1.

On Orthogonal Symmetric Chain Decompositions

“…Another recent result that received considerable attention is the paper by Konev and Lisitsa [KL14] on the Erdős discrepancy conjecture. SAT solvers have also been used in the context of geometry, specifically for tackling Erdős-Szekeres type questions, see the papers by Balko and Valtr [BV17] and by Scheucher [Sch18]. Moreover, with their help researchers were able to find new coil-in-the-box Gray codes [ZKC08] and to compute pairs of orthogonal diagonal Latin squares [ZKS16].…”