Cosine polynomials of the form f(x) = cos(a_1 x) + cos(a_2 x) + … + cos(a_n x) appear extensively in number theory and combinatorics. An old problem of Ankeny and Chowla asks: if a_1 … a_n are distinct positive integers, how small must the minimum value of f(x) on [0, 2π] be? Concurrently with Benjamin Bedert, we show that the minimum value of f(x) must be polynomial in n. Our proof is based on a new theorem in spectral graph theory: if G is a graph with average d and adjacency least eigenvalue at least -d^γ , then G must contain a clique of size d^{1 - O(γ)} . Joint work with Zhihan Jin, Aleksa Milojević, and István Tomon, with thanks to Ilya Shkredov.