A d-uniform hypergraph H is a sum hypergraph iff there is a finite S ⊆ IN such that H is isomorphic to the hypergraph H+d (S) = (V, E), where V = S and E = {{v1, . . . , vd} : (i 6= j ⇒ vi 6= vj)∧ ∑d i=1 vi ∈ S}. For an arbitrary d-uniform hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices w1, . . . , wσ 6∈ V such that H ∪ {w1, . . . , wσ} is a sum hyp...

The hypergraph of the Fano plane is the unique 3-uniform hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. This hypergraph is not 2-colorable, but becomes so on deleting any hyperedge from it. We show that taking uniformly at random a labeled 3-uniform hypergraph H on n vertices not containing the hypergraph of the Fano plane, H turns out t...

Let H = (V, E) be a 3-uniform linear hypergraph with one hypercycle C3. We consider a blow-up hypergraph B[H]. We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph B[H] of the hypergraph H, with hyperedge densities satisfying special conditions, such that the hypergraph H appears in a blow-up hypergraph as a transversal. We present an efficient...

Let F be an r-uniform hypergraph and G be a multigraph. The hypergraph F is a Berge-G if there is a bijection f : E(G) → E(F) such that e ⊆ f(e) for each e ∈ E(G). Given a family of multigraphs G, a hypergraph H is said to be G-free if for each G ∈ G, H does not contain a subhypergraph that is isomorphic to a Berge-G. We prove bounds on the maximum number of edges in an r-uniform linear hypergr...

Let k ≥ 2 be a fixed integer and P be a property of k-uniform hypergraphs. In other words, P is a (typically infinite) family of k-uniform hypergraphs and we say a given hypergraph H satisfies P if H ∈ P . For a given constant η > 0 a k-uniform hypergraph H on n vertices is η-far from P if no hypergraph obtained from H by changing (adding or deleting) at most ηn edges in H satisfies P . More pr...

Fix integers t ≥ r ≥ 2 and an r-uniform hypergraph F . We prove that the maximum number of edges in a t-partite r-uniform hypergraph on n vertices that contains no copy of F is ct,F ( n r ) + o(nr), where ct,F can be determined by a finite computation. We explicitly define a sequence F1, F2, . . . of r-uniform hypergraphs, and prove that the maximum number of edges in a t-chromatic r-uniform hy...

Let H be a 3-partite 3-uniform hypergraph with each partition class of size n, that is, a 3-uniform hypergraph such that every edge intersects every partition class in exactly one vertex. We determine the Dirac-type vertex degree thresholds for perfect matchings in 3-partite 3-uniform hypergraphs.

We study the following Maker/Breaker game. Maker and Breaker take turns in choosing vertices from a given n-uniform hypergraph F , with Maker going first. Maker’s goal is to completely occupy a hyperedge and Breaker tries to avoid this. Beck conjectures that if the maximum neighborhood size of F is at most 2 then Breaker has a winning strategy. We disprove this conjecture by establishing an n-u...

In 1986, Johnson and Perry proved a class of inequalities for uniform hypergraphs which included the following: for any such hypergraph, the geometric mean over the hyperedges of the geometric means of the degrees of the nodes on the hyperedge is no less than the average degree in the hypergraph, with equality only if the hypergraph is regular. Here, we prove a wider class of inequalities in a ...

We prove that the vertex degree threshold for tiling C3 4 (the 3-uniform hypergraph with four vertices and two triples) in a 3-uniform hypergraph on n ∈ 4N vertices is (n−1 2 ) − ( 4 n 2 ) + 8n + c, where c = 1 if n ∈ 8N and c = −2 otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling. C © 2014 Wiley Periodicals, Inc. J. Graph...