New bounds for a hypergraph bipartite Turán problem

• Published in: Journal of combinatorial theory. Series A Vol. 176; p. 105299
• Main Authors: Ergemlidze, Beka, Jiang, Tao, Methuku, Abhishek
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2020
• Subjects: Bipartite Turán; Degenerate problem; Extremal hypergraphs; Hypergraph Turán problem; Extremal hypergraphs; Bipartite Turán; Degenerate problem; Hypergraph Turán problem
• ISSN: 0097-3165
• DOI: 10.1016/j.jcta.2020.105299

Let t be an integer such that t≥2. Let K2,t(3) denote the triple system consisting of the 2t triples {a,xi,yi}, {b,xi,yi} for 1≤i≤t, where the elements a,b,x1,x2,…,xt, y1,y2,…,yt are all distinct. Let ex(n,K2,t(3)) denote the maximum size of a triple system on n elements that does not contain K2,t(3). This function was studied by Mubayi and Verstraëte [9], where the special case t=2 was a problem of Erdős [1] that was studied by various authors [3,9,10]. Mubayi and Verstraëte proved that ex(n,K2,t(3))<t4(n2) and that for infinitely many n, ex(n,K2,t(3))≥2t−13(n2). These bounds together with a standard argument show that g(t):=limn→∞⁡ex(n,K2,t(3))/(n2) exists and that2t−13≤g(t)≤t4. Addressing the question of Mubayi and Verstraëte on the growth rate of g(t), we prove that as t→∞,g(t)=Θ(t1+o(1)).