Weakly saturated hypergraphs and a conjecture of Tuza

Abstract

Given a fixed hypergraph H H , let wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) denote the smallest number of edges in an n n -vertex hypergraph G G , with the property that one can sequentially add the edges missing from G G , so that whenever an edge is added, a new copy of H H is created. The study of wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) was introduced by Bollobás in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most H H very little is known regarding wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) , Alon proved in 1985 that for every graph H H there is a limiting constant C H C_H so that wsat ⁡ ( n , H ) = ( C H + o ( 1 ) ) n \operatorname {wsat}(n,H)=(C_H+o(1))n . Tuza conjectured in 1992 that Alon’s theorem can be (appropriately) extended to arbitrary r r -uniform hypergraphs. In this paper we prove this conjecture.