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26077 results in Abstract analysis

Page 252 of 1304

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

  • Part of
  • CLAYTON COLLIER-CARTAINO, NATHAN GRABER, TAO JIANG
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 3 / May 2018
    Published online by Cambridge University Press:
    02 November 2017, pp. 358-386

  • 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 exL(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 Crℓ denote the r-uniform linear cycle of length ℓ, which is an r-graph with edges e1, . . ., eℓ such that, for all i ∈ [ℓ−1], |ei ∩ ei+1|=1, |eℓ ∩ e1|=1 and ei ∩ ej = ∅ for all other pairs {i,j}, i ≠ j. For all r ≥ 3 and ℓ ≥ 3, we show that there exists a positive constant c = cr,ℓ, depending only r and ℓ, such that exL(n,Crℓ) ≤ cn1+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 = am,r and b = bm,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}\quadR(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}.\end{equation}

Page 252 of 1304