Hypergraph Turán numbers of linear cycles

A k-uniform linear cycle of length ℓ, denoted by Cℓ(k), is a cyclic list of k-sets A1, . . . , Aℓ such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k ≥ 5 and ℓ ≥ 3 and sufficiently large n we determine the largest size of a k-uniform set family on [n] not containing a linear cycle of length ℓ. For odd ℓ = 2t + 1 the unique extremal family FS consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even ℓ = 2t + 2, the unique extremal family consists of FS plus all the k-sets outside S containing some fixed two elements. For k ≥ 4 and large n we also establish an exact result for so-called minimal cycles. For all k ≥ 4 our results substantially extend Erdos's result on largest k-uniform families without t + 1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstraëte. Our main method is the delta system method. © 2014 Elsevier Inc

Balanced supersaturation for some degenerate hypergraphs

A classical theorem of Simonovits from the 1980s asserts that every graph $G$ satisfying ${e(G) \gg v(G)^{1+1/k}}$ must contain $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$ copies of $C_{2k}$. Recently, Morris and Saxton established a balanced version of Simonovits' theorem, showing that such $G$ has $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$ copies of $C_{2k}$, which are `uniformly distributed' over the edges of $G$. Moreover, they used this result to obtain a sharp bound on the number of $C_{2k}$-free graphs via the container method. In this paper, we generalise Morris-Saxton's results for even cycles to $\Theta$-graphs. We also prove analogous results for complete $r$-partite $r$-graphs.Comment: Changed title, abstract and introduction were rewritte

Book free 33-Uniform Hypergraphs

A $k$-book in a hypergraph consists of $k$ Berge triangles sharing a common edge. In this paper we prove that the number of the hyperedges in a $k$-book-free 3-uniform hypergraph on $n$ vertices is at most $\frac{n^2}{8}(1+o(1))$

The Turán Density of Tight Cycles in Three-Uniform Hypergraphs

The Turán density of an $r$-uniform hypergraph ${\mathcal {H}}$, denoted $\pi ({\mathcal {H}})$, is the limit of the maximum density of an $n$-vertex $r$-uniform hypergraph not containing a copy of ${\mathcal {H}}$, as $n \to \infty$. Denote by ${\mathcal {C}}_{\ell }$ the $3$-uniform tight cycle on $\ell$ vertices. Mubayi and Rödl gave an “iterated blow-up” construction showing that the Turán density of ${\mathcal {C}}_{5}$ is at least $2\sqrt {3} - 3 \approx 0.464$, and this bound is conjectured to be tight. Their construction also does not contain ${\mathcal {C}}_{\ell }$ for larger $\ell$ not divisible by $3$, which suggests that it might be the extremal construction for these hypergraphs as well. Here, we determine the Turán density of ${\mathcal {C}}_{\ell }$ for all large $\ell$ not divisible by $3$, showing that indeed $\pi ({\mathcal {C}}_{\ell }) = 2\sqrt {3} - 3$. To our knowledge, this is the first example of a Turán density being determined where the extremal construction is an iterated blow-up construction. A key component in our proof, which may be of independent interest, is a $3$-uniform analogue of the statement “a graph is bipartite if and only if it does not contain an odd cycle”

On multicolor Ramsey numbers of triple system paths of length 3

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $\binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $\mathcal{L}$ be the loose 3-uniform path with 3 edges and $\mathcal{M}$ denote the messy 3-uniform path with 3 edges; that is, let $\mathcal{L} = \{abc, cde, efg\}$ and $\mathcal{M} = \{ abc, bcd, def\}$. In this note we prove $r_k(\mathcal{L}) < 1.55k$ and $r_k(\mathcal{M}) < 1.6k$ for $k$ sufficiently large. The former result improves on the bound $r_k( \mathcal{L}) < 1.975k + 7\sqrt{k}$, which was recently established by {\L}uczak and Polcyn.Comment: 18 pages, 3 figure