A generalization of the Erdös–Ko–Rado theorem

AbstractIn this note, we investigate some properties of local Kneser graphs defined in [János Körner, Concetta Pilotto, Gábor Simonyi, Local chromatic number and sperner capacity, J. Combin. Theory Ser. B 95 (1) (2005) 101–117]. In this regard, as a generalization of the Erdös–Ko–Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next, we provide an upper bound for their chromatic number

Cameron-Liebler sets of k-spaces in PG(n,q)

Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F. Ihringer. We list several equivalent definitions for these Cameron-Liebler sets, by making a generalization of known results about Cameron-Liebler line sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also present a classification result

The Distortion of Locality Sensitive Hashing

Given a pairwise similarity notion between objects, locality sensitive hashing (LSH) aims to construct a hash function family over the universe of objects such that the probability two objects hash to the same value is their similarity. LSH is a powerful algorithmic tool for large-scale applications and much work has been done to understand LSHable similarities, i.e., similarities that admit an LSH. In this paper we focus on similarities that are provably non-LSHable and propose a notion of distortion to capture the approximation of such a similarity by a similarity that is LSHable. We consider several well-known non-LSHable similarities and show tight upper and lower bounds on their distortion. We also experimentally show that our upper bounds translate to

High dimensional Hoffman bound and applications in extremal combinatorics

One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound is sharp on the tensor power of a graph whenever it is sharp for the original graph. In this paper, we introduce the related problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many of the prominent open problems in extremal combinatorics, such as the Tur\'an problem for (hyper-)graphs, can be encoded as special cases of this problem. We also give a new generalization of the Hoffman bound for hypergraphs which is sharp for the tensor power of a hypergraph whenever it is sharp for the original hypergraph. As an application of our Hoffman bound, we make progress on the problem of Frankl on families of sets without extended triangles from 1990. We show that if $\frac{1}{2}n\le2k\le\frac{2}{3}n,$ then the extremal family is the star, i.e. the family of all sets that contains a given element. This covers the entire range in which the star is extremal. As another application, we provide spectral proofs for Mantel's theorem on triangle-free graphs and for Frankl-Tokushige theorem on $k$-wise intersecting families

Cameron-Liebler sets of k-spaces in PG(n,q)

Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F. Ihringer. We list several equivalent definitions for these Cameron-Liebler sets, by making a generalization of known results about Cameron-Liebler line sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also present a classification result

Cameron-Liebler sets of k-spaces in PG(n,q)

Cameron-Liebler sets of k-spaces were introduced recently in Filmus and Ihringer (J Combin Theory Ser A, 2019). We list several equivalent definitions for these Cameron-Liebler sets, by making a generalization of known results about Cameron-Liebler line sets in PG(n,q) and Cameron-Liebler sets of k-spaces in PG(2k+1,q). We also present some classification results