Abstract. For a positive integer m and a subgroup Λ of the unit group (Z/mZ) × , the corresponding generalized Kloosterman sum is the function). Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.
If 2 ≤ d ≤ k and n ≥ dk/(d − 1), a d-cluster is defined to be a collection of d elements of [n] kwith empty intersection and union of size no more than 2k. Mubayi [6] conjectured that the largest size of a d-cluster-free family F ⊂ [n] k is n−1 k−1 , with equality holding only for a maximum-sized star. Here we prove two results. The first resolves Mubayi's conjecture and proves a stronger result, thus completing a new generalization of the Erdős-Ko-Rado Theorem. The second shows, by a different technique, that for a slightly more limited set of parameters only a very specific kind of d-cluster need be forbidden to achieve the same bound.
A d-simplex is defined to be a collection A 1 , . . . , A d+1 of subsets of size k of [n] such that the intersection of all of them is empty, but the intersection of any d of them is non-empty. Furthemore, a d-cluster is a collection of d + 1 such sets with empty intersection and union of size ≤ 2k, and a d-simplex-cluster is such a collection that is both a d-simplex and a dcluster. The Erdős-Chvátal d-simplex Conjecture from 1974 states that any family of k-subsets of [n] containing no d-simplex must be of size no greater than n−1 k−1 . In 2011, Keevash and Mubayi extended this conjecture by hypothesizing that the same bound would hold for families containing no d-simplex-cluster. In this paper, we resolve Keevash and Mubayi's conjecture for all 4 ≤ d + 1 ≤ k ≤ n/2, which in turn resolves all remaining cases of the Erdős-Chvátal Conjecture except when n is very small (i.e. n < 2k).
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