Turán problems and shadows I: Paths and cycles

Abstract

A k-path is a hypergraph P k = {e 1 , e 2 , . . . , e k } such that |e i ∩ e j | = 1 if |j − i| = 1 and e i ∩ e j = ∅ otherwise. A k-cycle is a hypergraph C k = {e 1 , e 2 , . . . , e k } obtained from a (k − 1)-path {e 1 , e 2 , . . . , e k−1 } by adding an edge e k that shares one vertex with e 1 , another vertex with e k−1 and is disjoint from the other edges.Let ex r (n, G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We prove that for fixed r ≥ 3, k ≥ 4 and (k, r) = (4, 3), for large enough n:if k is even and we characterize all the extremal r-graphs. We also solve the case (k, r) = (4, 3), which needs a special treatment. The case k = 3 was settled by Frankl and Füredi.This work is the next step in a long line of research beginning with conjectures of Erdős and Sós from the early 1970's. In particular, we extend the work (and settle a conjecture) of Füredi, Jiang and Seiver who solved this problem for P k when r ≥ 4 and of Füredi and Jiang who solved it for C k when r ≥ 5. They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.

On the number of linear hypergraphs of large girth

“…For integers and , an ‐uniform linear cycle of length , denoted by , is an ‐graph with edges such that for every , and for all other pairs . Kostochka, Mubayi, and Verstraëte , and independently, Füredi and Jiang proved that for every . Then by , we trivially have …”

“…Recently, the case when H is a linear cycle received more attention. For integers r 2 ≥ and 3 ℓ ≥ , an r-uniform linear cycle of length ℓ, denoted by C r ℓ , is an r-graph with edges e e , …, [17], and independently, Füredi and Jiang [10] proved that for every r nC n , 3, ex ( , ) = Θ( )…”

Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs

“…Theorem 2.2 [12,20,31,33,32] For sufficiently large n, we have 1. ex (n, s, P 2 ) = n−2 s−2 for s ≥ 4, and ex (n, 3, P 2 ) ≤ n.…”

“…Theorem 2.3 [19,33] Let s, t be positive integers with s ≥ 3. For sufficiently large n, we have ex (n, s, C 2t+1 ) = n s − n − t s and for (s, t) = (3, 1),…”

“…For the exceptional case of C 4 in 3-uniform hypergraphs, Kostochka, Mubayi and Verstraëte [33] showed that ex (n, 3,…”