TY - JOUR

T1 - Asymptotics of the Hypergraph Bipartite Turán Problem

AU - Bradač, Domagoj

AU - Gishboliner, Lior

AU - Janzer, Oliver

AU - Sudakov, Benny

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023/6

Y1 - 2023/6

N2 - For positive integers s, t, r, let Ks,t(r) denote the r-uniform hypergraph whose vertex set is the union of pairwise disjoint sets X, Y1, ⋯ , Yt , where | X| = s and | Y1| = ⋯ = | Yt| = r- 1 , and whose edge set is { { x} ∪ Yi: x∈ X, 1 ≤ i≤ t} . The study of the Turán function of Ks,t(r) received considerable interest in recent years. Our main results are as follows. First, we show that ex(n,Ks,t(r))=Os,r(t1s-1nr-1s-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,K2,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,Ks,t(3))=Os,t(n3-1s-1-εs) (for all s≥ 3). This indicates that the behaviour of ex(n,Ks,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.

AB - For positive integers s, t, r, let Ks,t(r) denote the r-uniform hypergraph whose vertex set is the union of pairwise disjoint sets X, Y1, ⋯ , Yt , where | X| = s and | Y1| = ⋯ = | Yt| = r- 1 , and whose edge set is { { x} ∪ Yi: x∈ X, 1 ≤ i≤ t} . The study of the Turán function of Ks,t(r) received considerable interest in recent years. Our main results are as follows. First, we show that ex(n,Ks,t(r))=Os,r(t1s-1nr-1s-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,K2,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,Ks,t(3))=Os,t(n3-1s-1-εs) (for all s≥ 3). This indicates that the behaviour of ex(n,Ks,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.

KW - Turan problem

KW - bipartite graphs

KW - hypergraphs

UR - http://www.scopus.com/inward/record.url?scp=85160225791&partnerID=8YFLogxK

U2 - 10.1007/s00493-023-00019-6

DO - 10.1007/s00493-023-00019-6

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AN - SCOPUS:85160225791

SN - 0209-9683

VL - 43

SP - 429

EP - 446

JO - Combinatorica

JF - Combinatorica

IS - 3

ER -