2-cancellative hypergraphs and codes

TL;DR

This paper investigates the properties of t-cancellative hypergraphs and codes, providing improved bounds on their sizes and an algebraic construction demonstrating their asymptotic behavior.

Contribution

It offers significantly improved upper bounds for the size of t-cancellative families and introduces an algebraic construction showing the asymptotic order of certain hypergraph families.

Findings

c(n,2) < 2^{0.322n} for large n

Order of magnitude of c_{2k}(n,2) is n^k as n grows

New algebraic construction for t-cancellative hypergraphs

Abstract

A family of sets F (and the corresponding family of 0-1 vectors) is called t-cancellative if for all distict t+2 members A_1,... A_t and B,C from F the union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let c(n,t) be the size of the largest t-cancellative family on n elements, and let c_k(n,t) denote the largest k-uniform family. We significantly improve the previous upper bounds, e.g., we show c(n,2)< 2^0.322n (for n> n_0). Using an algebraic construction we show that the order of magnitude of c_{2k}(n,2) is n^k for each k (when n goes to infinity).