Let m(k) denote the maximum number of edges in a non-extendable, intersecting k-graph. Erdős and Lovász proved that m(k) ≤ kk. For k ≥ 625 we prove m(k) < kk・e−k1/4/6.

In this paper we consider j-tuple-connected components in random k-uniform hypergraphs (the j-tuple-connectedness relation can be defined by letting two j-sets be connected if they lie in a common edge and considering the transitive closure; the case j = 1 corresponds to the common notion of vertex-connectedness). We show that the existence of a j-tuple-connected component containing Θ(nj) j-sets undergoes a phase transition and show that the threshold occurs at edge probability

$$\frac{(k-j)!}{\binom{k}{j}-1}n^{j-k}.$$

Our main original contribution is a bounded degree lemma, which controls the structure of the component grown in the search process.

It follows from known results that every regular tripartite hypergraph of positive degree, with n vertices in each class, has matching number at least n/2. This bound is best possible, and the extremal configuration is unique. Here we prove a stability version of this statement, establishing that every regular tripartite hypergraph with matching number at most (1 + ϵ)n/2 is close in structure to the extremal configuration, where ‘closeness’ is measured by an explicit function of ϵ.

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if, for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.

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}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

Given a family of r-uniform hypergraphs ${\cal F}$ (or r-graphs for brevity), the Turán number ex(n,${\cal F})$ of ${\cal F}$ is the maximum number of edges in an r-graph on n vertices that does not contain any member of ${\cal F}$. A pair {u,v} is covered in a hypergraph G if some edge of G contains {u, v}. Given an r-graph F and a positive integer p ⩾ n(F), where n(F) denotes the number of vertices in F, let HFp denote the r-graph obtained as follows. Label the vertices of F as v1,. . .,vn(F). Add new vertices vn(F)+1,. . .,vp. For each pair of vertices vi, vj not covered in F, add a set Bi,j of r − 2 new vertices and the edge {vi, vj} ∪ Bi,j, where the Bi,j are pairwise disjoint over all such pairs {i, j}. We call HFp the expanded p-clique with an embedded F. For a relatively large family of F, we show that for all sufficiently large n, ex(n,HFp) = |Tr(n, p − 1)|, where Tr(n, p − 1) is the balanced complete (p − 1)-partite r-graph on n vertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Turán number is exactly determined (for large n).

The 3-uniform tight cycle Cs3 has vertex set ${\mathbb Z}_s$ and edge set {{i, i + 1, i + 2}: i ∈ ${\mathbb Z}_s$}. We prove that for every s ≢ 0 (mod 3) with s ⩾ 16 or s ∈ {8, 11, 14} there is a cs > 0 such that the 3-uniform hypergraph Ramsey number r(Cs3, Kn3) satisfies

$$\begin{equation*} r(C_s^3, K_n^3)< 2^{c_s n \log n}.\ \end{equation*}$$

$2^{n^{1+\epsilon_s}}$

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ⩾ 4 and 0 < p < 1. Suppose that H is an n-vertex triple system with r|n and the following two properties:

• • for every graph G with V(G) = V(H), at least p proportion of the triangles in G are also edges of H,

• • for every vertex x of H, the link graph of x is a quasirandom graph with density at least p.

We give an easy method for constructing containers for simple hypergraphs. The method also has consequences for non-simple hypergraphs. Some applications are given; in particular, a very transparent calculation is offered for the number of H-free hypergraphs, where H is some fixed uniform hypergraph.

We establish a relation between two uniform models of random k-graphs (for constant k ⩾ 3) on n labelled vertices: ℍ(k)(n,m), the random k-graph with exactly m edges, and ℍ(k)(n,d), the random d-regular k-graph. By extending the switching technique of McKay and Wormald to k-graphs, we show that, for some range of d = d(n) and a constant c > 0, if m ~ cnd, then one can couple ℍ(k)(n,m) and ℍ(k)(n,d) so that the latter contains the former with probability tending to one as n → ∞. In view of known results on the existence of a loose Hamilton cycle in ℍ(k)(n,m), we conclude that ℍ(k)(n,d) contains a loose Hamilton cycle when d ≫ log n (or just d ⩾ C log n, if k = 3) and d = o(n1/2).

A perfect Kt-matching in a graph G is a spanning subgraph consisting of vertex-disjoint copies of Kt. A classic theorem of Hajnal and Szemerédi states that if G is a graph of order n with minimum degree δ(G) ≥ (t − 1)n/t and t|n, then G contains a perfect Kt-matching. Let G be a t-partite graph with vertex classes V1, …, Vt each of size n. We show that, for any γ > 0, if every vertex x ∈ Vi is joined to at least $\bigl ((t-1)/t + \gamma \bigr )n$ vertices of Vj for each j ≠ i, then G contains a perfect Kt-matching, provided n is large enough. Thus, we verify a conjecture of Fischer [6] asymptotically. Furthermore, we consider a generalization to hypergraphs in terms of the codegree.

A 3-graph is said to contain a generalized 4-cycle if it contains 4 edges A, B, C, D such that A ∩ B=C ∩ D =∅ and A ∪ B=C ∪ D. We show that a 3-graph in which every pair of vertices is contained in at least 4 edges must contain a generalized 4-cycle. When the number of vertices, n, is equivalent to 1 or 5 modulo 20, this result is optimum, in the sense that for such n there are 3-graphs where every pair of vertices is contained in 3 edges but which do not contain a generalized 4-cycle.

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices is j-independent if its intersection with any edge has size strictly less than j. The Kruskal–Katona theorem implies that in an r-uniform hypergraph with a fixed size and order, the hypergraph with the most r-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number of j-independent sets in an r-uniform hypergraph.