Asymptotics of the Hypergraph Bipartite Turán Problem

• Published in: Combinatorica (Budapest. 1981) Vol. 43; no. 3; pp. 429 - 446
• Main Authors: Bradač, Domagoj, Gishboliner, Lior, Janzer, Oliver, Sudakov, Benny
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2023; Springer Nature B.V
• Subjects: Combinatorics; Graph theory; Graphs; Mathematics; Mathematics and Statistics; Original Paper; Vertex sets; 05C65; Turan problem; bipartite graphs; hypergraphs; 05C35
• ISSN: 0209-9683; 1439-6912
• DOI: 10.1007/s00493-023-00019-6

For positive integers s ,  t ,  r , let K s , t ( r ) denote the r -uniform hypergraph whose vertex set is the union of pairwise disjoint sets X , Y 1 , ⋯ , Y t , where | X | = s and | Y 1 | = ⋯ = | Y t | = r - 1 , and whose edge set is { { x } ∪ Y i : x ∈ X , 1 ≤ i ≤ t } . The study of the Turán function of K s , t ( r ) received considerable interest in recent years. Our main results are as follows. First, we show that 1 ex n , K s , t ( r ) = O s , r t 1 s - 1 n r - 1 s - 1 for all s , t ≥ 2 and r ≥ 3 , improving the power of n in the previously best bound and resolving a question of Mubayi and Verstraëte about the dependence of ex ( n , K 2 , t ( 3 ) ) on t . Second, we show that ( 1 ) is tight when r is even and t ≫ s . This disproves a conjecture of Xu, Zhang and Ge. Third, we show that ( 1 ) is not tight for r = 3 , namely that ex ( n , K s , t ( 3 ) ) = O s , t ( n 3 - 1 s - 1 - ε s ) (for all s ≥ 3 ). This indicates that the behaviour of ex ( n , K s , t ( r ) ) might depend on the parity of r . Lastly, we prove a conjecture of Ergemlidze, Jiang and Methuku on the hypergraph analogue of the bipartite Turán problem for graphs with bounded degrees on one side. Our tools include a novel twist on the dependent random choice method as well as a variant of the celebrated norm graphs constructed by Kollár, Rónyai and Szabó.