By a tight tour in a k-uniform hypergraph H we mean any sequence of its vertices (w0, w1, . . . , ws−1) such that for all i = 0, . . . , s−1 the set ei = {wi, wi+1 . . . , wi+k−1} is an edge ofH (where operations on indices are computed modulo s) and the sets ei for i = 0, . . . , s − 1 are pairwise different. A tight tour in H is a tight Euler tour if it contains all edges ofH . We prove that ...

We will give a tight minimum co-degree condition for a 4-uniform hypergraph to contain a perfect matching.

We consider high-order connectivity in k-uniform hypergraphs defined as follows: Two j-sets are j-connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. We describe the evolution of jconnected components in the k-uniform binomial random hypergraph H(n, p). In particular, we determine the asymptotic size of the giant component shortly...

For a graph G = (V,E) and a non-empty set X, a linear hypergraph set-indexer (LHSI) is a function f : V (G)→ 2X satisfying the following conditions: (i)f is injective (ii) the ordered pair Hf (G) = (X, f(V )), where f(V ) = {f(v) : v ∈ V (G)}, is a linear hypergraph, (iii) the induced set-valued function f⊕ : E → 2X , defined by f⊕(uv) = f(u)⊕ f(v),∀ uv ∈ E is injective, and (iv) Hf⊕(G) = (X, f...

Let 2 ≤ q ≤ min{p, t − 1} be fixed and n → ∞. Suppose that F is a p-uniform hypergraph on n vertices that contains no complete q-uniform hypergraph on t vertices as a trace. We determine the asymptotic maximum size of F in many cases. For example, when q = 2 and p ∈ {t, t+ 1}, the maximum is ( n t−1 ) t−1 + o(nt−1), and when p = t = 3, it is b (n−1) 2 4 c for all n ≥ 3. Our proofs use the Krusk...

A conjecture of Erdős from 1965 suggests the minimum number of edges in a kuniform hypergraph on n vertices which forces a matching of size t, where t ≤ n/k. Our main result verifies this conjecture asymptotically, for all t < 0.48n/k. This gives an approximate answer to a question of Huang, Loh and Sudakov, who proved the conjecture for t ≤ n/3k. As a consequence of our result, we extend bound...

Let α1, . . . , αk satisfy ∑ i αi = 1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1, . . . , Ak of sizes α1n, . . . , αkn, the number of edges intersecting A1, . . . , Ak is (asymptotically) the number one would expect to find in a random k-uniform hypergraph. Can we then infer that H is quasi-random? We show t...

Let H(n,N), where k ≥ 2, be a random hypergraph on the vertex set [n] = {1, 2, . . . , n} with N edges drawn independently with replacement from all subsets of [n] of size k. For d̄ = kN/n and any ε > 0 we show that if k = o(ln(d̄/ lnn)) and k = o(ln(n/ ln d̄)), then with probability 1−o(1) a random greedy algorithm produces a proper edge-colouring of H(n,N) with at most d̄(1+ε) colours. This yield...

A classical result of Erdős and Hajnal claims that for any integers k, r, g ≥ 2 there is an r-uniform hypergraph of girth at least g with chromatic number at least k. This implies that there are sparse hypergraphs such that in any coloring of their vertices with at most k − 1 colors there is a monochromatic hyperedge. We show that for any integers r, g ≥ 2 there is an r-uniform hypergraph of gi...

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees of the complete graph on (n + 1) vertices and Parking functions of length n, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-P...