Weakly saturated hypergraphs and a conjecture of Tuza

Given a fixed hypergraph H H , let alttext="w s t left-parenthesis n comma upper H right-parenthesis"> wsat ⁡ ( n , stretchy="false">) encoding="application/x-tex">\operatorname {wsat}(n,H) denote the smallest number of edges in an alttext="n"> encoding="application/x-tex">n -vertex G"> G encoding="application/x-tex">G with property that one can sequentially add missing from so whenever edge is added, new copy created. The study was introduced by Bollobás 1968, and turned out to be most influential topics extremal combinatorics. While for very little known regarding Alon proved 1985 every graph there limiting constant C Subscript C encoding="application/x-tex">C_H right-parenthesis equals Baseline plus o 1 n"> = + o 1 {wsat}(n,H)=(C_H+o(1))n . Tuza conjectured 1992 Alon’s theorem (appropriately) extended arbitrary alttext="r"> r encoding="application/x-tex">r -uniform hypergraphs. In this paper we prove conjecture.