A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1

For $n \geq 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^n$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^n$ times the volume of the ball. This disproves a conjecture of Larman and Rogers from 1972.Comment: 3 pages, 1 figure; final version to appear in Mathematik

Complete positivity and distance-avoiding sets

We introduce the cone of completely-positive functions, a subset of the cone of positive-type functions, and use it to fully characterize maximum-density distance-avoiding sets as the optimal solutions of a convex optimization problem. As a consequence of this characterization, it is possible to reprove and improve many results concerning distance-avoiding sets on the sphere and in Euclidean space.Comment: 57 pages; minor corrections in comparison to the previous versio

kk-point semidefinite programming bounds for equiangular lines

We give a hierarchy of $k$-point bounds extending the Delsarte-Goethals-Seidel linear programming $2$-point bound and the Bachoc-Vallentin semidefinite programming $3$-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute~$4$, $5$, and $6$-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.Comment: 26 pages, 4 figures. New introduction and references update

A recursive Lov\'asz theta number for simplex-avoiding sets

We recursively extend the Lov\'asz theta number to geometric hypergraphs on the unit sphere and on Euclidean space, obtaining an upper bound for the independence ratio of these hypergraphs. As an application we reprove a result in Euclidean Ramsey theory in the measurable setting, namely that every $k$-simplex is exponentially Ramsey, and we improve existing bounds for the base of the exponential.Comment: 13 pages, 3 figure

A recursive theta body for hypergraphs

The theta body of a graph, introduced by Gr\"otschel, Lov\'asz, and Schrijver in 1986, is a tractable relaxation of the independent-set polytope derived from the Lov\'asz theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the high-dimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: triangle-free graphs and Mantel's theorem, and bounds on the density of triangle-avoiding sets in the Hamming cube.Comment: 23 pages, 2 figure