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Linear Turán Numbers of Linear Cycles and Cycle-Complete Ramsey Numbers


An r-uniform hypergraph is called an r-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear r-graph H and a positive integer n, the linear Turán number ex L (n,H) is the maximum number of edges in a linear r-graph G that does not contain H as a subgraph. For each ℓ ≥ 3, let C r denote the r-uniform linear cycle of length ℓ, which is an r-graph with edges e 1, . . ., e such that, for all i ∈ [ℓ−1], |e i e i+1|=1, |e e 1|=1 and e i e j = ∅ for all other pairs {i,j}, ij. For all r ≥ 3 and ℓ ≥ 3, we show that there exists a positive constant c = c r,ℓ, depending only r and ℓ, such that ex L (n,C r ) ≤ cn 1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.

Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constants a = a m,r and b = b m,r , depending only on m and r, such that \begin{equation} R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}} \quad\text{and}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

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Combinatorics, Probability and Computing
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