Let p(k) denote the partition function of k. For each k ≥ 2, we describe a list of p(k) − 1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on hypergraph quasirandomness, beginning with the early work of Chung and Graham and Frankl-Rödl related to strong hypergraph regularity, the spectral approach of Friedman-Wigderson, and more recent results of...

In this paper, we study a k-uniform directed hypergraph in general form and introduce its associated tensors. We present different spectral properties show that some of them are generalization the classical results for undirected hypergraphs. The notation odd-bipartite presented characterizations comparing with ones hypergraphs studied. also power cored investigate their properties.

We prove that for every positive integer m there is a finite point set $$\cal{P}$$ in the plane such no matter how three-colored, always disk containing exactly points, all of same color. This improves result Pach, Tardos and Tóth who proved two colors. The main ingredient construction subconstruction whose points are convex position. Namely, we show position two-colored, also unit disks simila...

For k-uniform hypergraphs F and H and an integer r ≥ 2, let cr,F (H) denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F (n) = maxH∈Hn cr,F (H), where the maximum is over the family Hn of all k-uniform hypergraphs on n vertices. Moreover, let ex(n, F ) be the usual extremal function, i.e., the maximum number of hyperedges of an n-vertex k-...

We prove a hypergraph analogue of the Erdős-Gallai theorem: for any δ > 0 there exists n0 such that for any 0 ≤ α ≤ 1, any k-uniform hypergraph H on n ≥ n0 vertices with at least (α+ δ) ( n k ) edges contains a tight cycle of length at least αn. Moreover, H contains tight cycles of all lengths up to αn which are divisible by k. A key tool in our proof is a ‘regular slice lemma’, which is derive...

Fix r ≥ 2 and a collection of r-uniform hypergraphs H. What is the minimum number of edges in an H-free r-uniform hypergraph with chromatic number greater than k? We investigate this question for various H. Our results include the following: • An (r, l)-system is an r-uniform hypergraph with every two edges sharing at most l vertices. For k sufficiently large, there is an (r, l)-system with chr...

A substantial amount of research in graph theory continues to concentrate on the existence of hamiltonian cycles and perfect matchings. A classic theorem of Dirac states that a sufficient condition for an n-vertex graph to be hamiltonian, and thus, for n even, to have a perfect matching, is that the minimum degree is at least n/2. Moreover, there are obvious counterexamples showing that this is...

This work studies the hardness of finding independent sets in hypergraphs which are either 2colorable or are almost 2-colorable, i.e. can be 2-colored after removing a small fraction of vertices and the incident hyperedges. To be precise, say that a hypergraph is (1−ε)-almost 2-colorable if removing an ε fraction of its vertices and all hyperedges incident on them makes the remaining hypergraph...

We prove that in a random 3-uniform or 4-uniform hypergraph of order n the probability that some two vertices have the same degree tends to one as n→∞.