Random Turán Theorem for Expansions of Spanning Subgraphs of Tight Trees

Abstract. The [Formula: see text]-expansion of a [Formula: see text]-uniform hypergraph [Formula: see text], denoted by [Formula: see text], is an [Formula: see text]-uniform hypergraph obtained by enlarging each [Formula: see text]-edge of [Formula: see text] with a set of [Formula: see text] vertices of degree one. The random Turán number [Formula: see text] is the maximum number of edges in an [Formula: see text]-free subgraph of [Formula: see text], where [Formula: see text] is the Erdős–Rényi random [Formula: see text]-graph with parameter [Formula: see text]. In this paper, we prove an upper bound for [Formula: see text] when [Formula: see text] belongs to a large family of [Formula: see text]-partite [Formula: see text]-graphs: the [Formula: see text]-expansion of spanning subgraphs of tight trees. This upper bound is essentially tight for at least the following two families of hypergraphs: 1. Our upper bounds are essentially tight for expansions of [Formula: see text], the complete [Formula: see text]-graph on [Formula: see text] vertices. The proof of the lower bound makes use of a recent construction of Gowers and Janzer generalizing the famous Ruzsa–Szemerédi construction. In particular, when [Formula: see text], this answers a question of the current author, Spiro, and Verstraëte concerning the random Turán number of linear triangle. 2. Let [Formula: see text] be a tight tree such that the intersection of all edges of [Formula: see text] is empty. Simple construction shows that the upper bounds we have for expansions of [Formula: see text] are essentially tight. The main technical contribution of this paper is a new way to obtain balanced supersaturation results for expansions of hypergraphs: we combine two ideas, one of Mubayi and Yepremyan and another of Balogh, Narayanan, and Skokan, via codegree dichotomy. We note that neither of these two ideas alone would be enough to recover results in this paper.