We say that a k-uniform hypergraph C is a Hamilton cycle of type `, for some 1 ≤ ` ≤ k, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices and for every pair of consecutive edges Ei−1, Ei in C (in the natural ordering of the edges) we have |Ei−1 \ Ei| = `. We prove that for k/2 < ` ≤ k, with high probability almost all edges of the ran...

Fix l ≥ r ≥ 2. Let H l+1 be the r-uniform hypergraph obtained from the complete graph Kl+1 by enlarging each edge with a new set of r− 2 vertices. Thus H l+1 has (r− 2) ( l+1 2 ) + l +1 vertices and ( l+1 2 ) edges. We prove that the maximum number of edges in an n-vertex r-uniform hypergraph containing no copy of H l+1 is (l)r lr ( n r ) + o(n) as n → ∞. This is the first infinite family of ir...

We consider the problem of learning a hypergraph using edge-detecting queries. In this model, the learner may query whether a set of vertices induces an edge of the hidden hypergraph or not. We show that an r-uniform hypergraph with m edges and n vertices is learnable with O(24rm · poly(r, logn)) queries with high probability. The queries can be made in O(min(2r(logm + r)2, (logm + r)3)) rounds...

We consider the following definition of connectedness in k-uniform hypergraphs: two j-sets (sets of j vertices) are j-connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. The hypergraph is jconnected if all j-sets are pairwise j-connected. We determine the threshold at which the random k-uniform hypergraph with edge probability p b...

We investigate extremal problems for quasirandom hypergraphs. We say that a 3-uniform hypergraph H “ pV,Eq is pd, η, q-quasirandom if for any subset X Ď V and every set of pairs P Ď V ˆV the number of pairs px, py, zqq P XˆP with tx, y, zu being a hyperedge of H is in the interval d |X| |P | ̆ η |V |. We show that for any ε ą 0 there exists η ą 0 such that every sufficiently large p1{2` ε, η, q...

We say that a 3-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. Also, let C4 denote the 3-uniform hypergraph on 4 vertices which contains 2 edges. We prove that for every ε > 0 there is an n0 such that for every n n0 the following holds: Every 3-un...

We characterize the position value for arbitrary hypergraph communication situations. The position value is first presented by the Shapley value of the uniform hyperlink game or the k-augmented uniform hyperlink game, which are obtained from the given hypergraph communication situation. These results generalize the non-axiomatic characterization of the position value for communication situation...

Let H = (V,E) be an r-uniform hypergraph and let F ⊂ 2 . A matching M of H is (α,F)perfect if for each F ∈ F , at least α|F | vertices of F are covered by M . Our main result is a theorem giving sufficient conditions for an r-uniform hypergraph to have a (1− ,F)-perfect matching. As a special case of our theorem we obtain the following result. Let K(n, r) denote the complete r-uniform hypergrap...

A coloring of the vertices of a hypergraph H is called conflict-free if each edge e of H contains a vertex whose color does not get repeated in e. The smallest number of colors required for such a coloring is called the conflict-free chromatic number of H, and is denoted by χCF (H). Pach and Tardos studied this parameter for graphs and hypergraphs. Among other things, they proved that for an (2...