A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in R n was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle θ and sufficiently large n there are at most 2n − 2 lines in R n with common angle θ. Moreover, this bound is achieved if and only if θ = arccos . Indeed, we show that for all θ = arccos 1 3 and and sufficiently large n, the number of equiangular lines is at most 1.93n. We also show that for any set of k fixed angles, one can find at most O(n k ) lines in R n having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.

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The minrank of a graph G on the set of vertices [n] over a field F is the minimum possible rank of a matrix M ∈ F n×n with nonzero diagonal entries such that M i,j = 0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n, p) over any finite or infinite field, showing that for every field F = F(n) and every p = p(n) satisfying n −1 ≤ p ≤ 1 − n −0.99 , the minrank of G = G(n, p) over F is Θ( n log(1/p) log n ) with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev, and Weinstein, who proved the above result for finite fields of size at most n O(1) , with tools from linear algebra, including an estimate of Rónyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials.

Let F be a finite union-closed family of sets whose largest set contains n elements. In [6], Wójcik defined the density of F to be the ratio of the average set size of F to n and conjectured that the minimum density over all union-closed families whose largest set contains n elements is (1 + o(1)) log 2 n/(2n) as n → ∞. We use a result of Reimer [3] to show that the density of F is always at least log 2 n/(2n), verifying Wójcik's conjecture. As a corollary we show that for n ≥ 16, some element must appear in at least (log 2 n)/n(|F |/2) sets of F . Preliminaries and NotationsGiven a family of sets F , we say F is union-closed if for all A, B ∈ F , A ∪ B ∈ F . In what follows, a union-closed family will always be taken to mean a finite union-closed family of finite sets. Let N = {1, 2, 3, . . .} be the set of natural numbers. We denote the cardinality of a finite set A by |A| := x∈A 1 and denote the union of a family of sets F by F := A∈F A. Given sets A, B, the set difference A \ B := {x ∈ A|x / ∈ B}. Let F be a union-closed family and let n = | F |. To avoid trivial cases, in this paper we only consider F with n ≥ 1. Define F a = {S ∈ F |a ∈ S} for all a ∈ F . We define the density of F by

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