From small eigenvalues to large cuts, and Chowla's cosine problem

Abstract

We prove that every graph with average degree $d$ and smallest adjacency eigenvalue $|λ_n|\leq d^γ$ contains a clique of size $d^{1-O(γ)}$. A simple corollary of this yields the first polynomial bound for Chowla's cosine problem (1965): for every finite set $A\subseteq \mathbb{Z}_{>0}$, the minimum of the cosine polynomial satisfies $$\min_{x\in [0, 2π]}\sum_{a\in A}\cos(ax)\leq -|A|^{1/10-o(1)}.$$ Another application makes significant progress on the problem of MaxCut in $H$-free graphs initiated by Erdős and Lovász in the 1970's. We show that every $m$-edge graph with no clique of size $m^{1/2-δ}$ has a cut of size at least $m/2+m^{1/2+\varepsilon}$ for some $\varepsilon=\varepsilon(δ)>0$.

Theorems & Definitions (121)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 3.1
• proof
• Claim 3.2
• proof