Spectral Methods in Extremal Combinatorics

Extremal combinatorics studies how large a collection of objects can be if it satisfies a given set of restrictions. Inspired by a classical theorem due to Erdos, Ko and Rado, Simonovits and Sos posed the following problem: determine how large a collection of graphs on the vertex set {1,...,n} can be, if the intersection of any two of them contains a triangle. They conjectured that the largest possible collection, containing 1/8 of all graphs, consists of all graphs containing a fixed triangle (a triangle-star). The first major contribution of this thesis is a confirmation of this conjecture. We prove the Simonovits–Sos conjecture in the following strong form: the only triangle-intersecting families of measure at least 1/8 are triangle-stars (uniqueness), and every triangle-intersecting family of measure 1/8−e is O(e)-close to a triangle-star (stability). In order to prove the stability part of our theorem, we utilize a structure theorem for Boolean functions on {0,1}^m whose Fourier expansion is concentrated on the first t+1 levels, due to Kindler and Safra. The second major contribution of this thesis consists of two analogs of this theorem for Boolean functions on S_m whose Fourier expansion is concentrated on the first two levels. In the same way that the Kindler–Safra theorem is useful for studying triangle-intersecting families, our structure theorems are useful for studying intersecting families of permutations, which are families in which any two permutations agree on the image of at least one point. Using one of our theorems, we give a simple proof of the following result of Ellis, Friedgut and Pilpel: an intersecting family of permutations on S_m of size (1−e)(m−1)! is O(e)-close to a double coset, a family which consists of all permutations sending some point i to some point j.