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

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

Covering Arrays for Equivalence Classes of Words

Covering arrays for words of length t over a d letter alphabet are k × n arrays with entries from the alphabet so that for each choice of t columns, each of the dt t-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case, words are equivalent if they induce the same partition of a t element set. In the second case, words of the same weighted sum are equivalent. In both cases we produce logarithmic upper bounds on the minimum size k = k(n) of a covering array. Most definitive results are for t = 2, 3, 4

Algebraic shifting of cyclic polytopes and stacked polytopes

AbstractGil Kalai introduced the shifting-theoretic upper bound relation as a method to generalize the g-theorem for simplicial spheres by using algebraic shifting. We will study the connection between the shifting-theoretic upper bound relation and combinatorial shifting. Also, we will compute the exterior algebraic shifted complex of the boundary complex of the cyclic d-polytope as well as of a stacked d-polytope. It will turn out that, in both cases, the exterior algebraic shifted complex coincides with the symmetric algebraic shifted complex

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

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