One of the earliest results in Combinatorics is Mantel’s theorem from 1907 that the largest triangle-free graph on a given vertex set is complete bipartite. However, a seemingly similar question posed by Turán in 1941 is still open: what is the largest 3-uniform hypergraph on a given vertex set with no tetrahedron? This question can be considered a test case for the general hypergraph Turán pro...

In this paper we present a graph property with sensitivity Θ( √ n), where n = (v 2 ) is the number of variables, and generalize it to a k-uniform hypergraph property with sensitivity Θ( √ n), where n = (v k ) is again the number of variables. This yields the smallest sensitivity yet achieved for a k-uniform hypergraph property. We then show that, for even k, there is a k-uniform hypergraph prop...

In this paper we extend the study of bipartite graphs with the induced ε-density property introduced by Frankl, Rödl, and the author. For a given kpartite k-uniform hypergraph G we say that a k-partite k-uniform hypergraph R = (W1, . . . ,Wk,F) has the induced ε-density property if every subhypergraph ofR with at least ε|F| edges contains a copy of G which is an induced subhypergraph of R. We s...

For 1 ≤ ` < k, an `-overlapping cycle is a k-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of k consecutive vertices and every two consecutive edges share exactly ` vertices. A k-uniform hypergraph H is `-Hamiltonian saturated, 1 ≤ ` ≤ k − 1, if H does not contain an `-overlapping Hamiltonian cycle C (k) n (`) but every hypergraph obtained from H by adding on...

We determine the strong total chromatic number of the complete h-uniform hypergraph Kh, and the complete h-partite hypergraph K,

In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraph with 2(logn) Ω(1) colors. This result was proved using a novel outer PCP verifier which had a strong soundness guarantee. We reduce the arity in their result by modifying their 12-query inner verifier to an 8-query inner verifier based on the hypergraph coloring hardness r...

A seminal result of Rödl (the Rödl nibble) asserts that the edges of the complete r-uniform hypergraph Kr n can be packed, almost completely, with copies of Kr k , where k is fixed. We prove that the same result holds in a dense hypergraph setting. It is shown that for every r-uniform hypergraph H0, there exists a constant α = α(H0) < 1 such that every r-uniform hypergraph H in which every (r −...

We show that, for a natural notion of quasirandomness in k-uniform hypergraphs, any quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(nk−1) contains a loose Hamilton cycle. We also give a construction to show that a k-uniform hypergraph satisfying these conditions need not contain a Hamilton `-cycle if k − ` divides k. The remaining values of ...

The paper deals with a problem concerning equitable vertex colorings of uniform hypergraphs, i.e., under which there are no monochromatic edges and simultaneously all the color classes have almost same cardinalities. We obtain new bound on edge number an n-uniform hypergraph that guarantees existence coloring r colors for this hypergraph.