Explicit Ramsey graphs and orthonormal labelings

We describe an explicit construction of triangle-free graphs with no independent sets of size m and with Ω(m3/2) vertices, improving a sequence of previous constructions by various authors. As a byproduct we show that the maximum possible value of the Lovász θ-function of a graph on n vertices with no independent set of size $3$ is Θ(n1/3), slightly improving a result of Kashin and Konyagin who showed that this maximum is at least Ω(n1/3/ log n) and at most O(n1/3). Our results imply that the maximum possible Euclidean norm of a sum of n unit vectors in Rⁿ, so that among any three of them some two are orthogonal, is Θ(n2/3).