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A collection of distinct sets is called a sunflower if the intersection of any pair of sets equals the common intersection of all the sets. Sunflowers are fundamental objects in extremal set theory with relations and applications to many other areas of mathematics as well as to theoretical computer science. A central problem in the area due to Erdős and Rado from 1960 asks for the minimum number of sets of size r r needed to guarantee the existence of a sunflower of a given size. Despite a lot of recent attention including a polymath project and some amazing breakthroughs, even the asymptotic answer remains unknown. We study a related problem first posed by Duke and Erdős in 1977 which requires that in addition the intersection size of the desired sunflower be fixed. This question is perhaps even more natural from a graph theoretic perspective since it asks for the Turán number of a hypergraph made by the sunflower consisting of k k edges, each of size r r and with common intersection of size t t . For a fixed size of the sunflower k k , the order of magnitude of the answer has been determined by Frankl and Füredi. In the 1980’s, with certain applications in mind, Chung, Erdős and Graham considered what happens if one allows k k , the size of the desired sunflower, to grow with the size of the ground set. In the three uniform case, r = 3 r=3 , the correct dependence on the size of the sunflower has been determined by Duke and Erdős and independently by Frankl and in the four uniform case by Bucić, Draganić, Sudakov and Tran. We resolve this problem for any uniformity, by determining up to a constant factor the n n -vertex Turán number of a sunflower of arbitrary uniformity r r , common intersection size t t and with the size of the sunflower k k allowed to grow with n n .

For positive integers s, t, r, let K(r)s,t 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 K(r)s,t received considerable interest in recent years. Our main results are as follows. First, we show that ex(n,K(r)s,t)=Os,r(t1s−1nr−1s−1)(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(3)2,t) 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(3)s,t)=Os,t(n3−1s−1−εs) (for all s≥3). This indicates that the behaviour of ex(n,K(r)s,t) 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., Combinatorica, ISSN:0209-9683, ISSN:1439-6912