For a hypergraph H and a set S, the trace of H on S is the set of all intersections of edges of H with S. We will consider forbidden trace problems, in which we want to find the largest hypergraph H that does not contain some list of forbidden configurations as traces, possibly with some restriction on the number of vertices or the size of the edges in H. In this paper we will focus on combinat...

A. The regularity lemma for 3-uniform hypergraphs asserts that every large hypergraph can be decomposed into a bounded number of quasi-random structures consisting of a sub-hypergraph and a sparse underlying graph. In this paper we show that in such a quasirandom structure most pairs of the edges of the graph can be connected by hyperpaths of length at most twelve. Some applications are ...

Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. A number of algorithms exist in the literature that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received attention in the literature as compared to the extensive studies on the guarantees of graph p...

Let k ≥ 2 and F be a linear k-uniform hypergraph with v vertices. We prove that if n is sufficiently large and v|n, then every quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum degree Ω(nk−1) admits a perfect F -packing. The case k = 2 follows immediately from the blowup lemma of Komlós, Sárközy, and Szemerédi. We also prove positive results for some nonlinea...

Given positive integers a ≤ b ≤ c, let Ka,b,c be the complete 3-partite 3-uniform hypergraph with three parts of sizes a, b, c. Let H be a 3-uniform hypergraph on n vertices where n is divisible by a + b + c. We asymptotically determine the minimum vertex degree of H that guarantees a perfect Ka,b,ctiling, that is, a spanning subgraph of H consisting of vertex-disjoint copies of Ka,b,c. This pa...

A k-uniform hypergraph is hamiltonian if for some cyclic ordering of its vertex set, every k consecutive vertices form an edge. In 1952 Dirac proved that if the minimum degree in an n-vertex graph is at least n/2 then the graph is hamiltonian. We prove an approximate version of an analogous result for uniform hypergraphs: For every k ≥ 3 and γ > 0, and for all n large enough, a sufficient condi...

We show that the Ramsey number is linear for every uniform hypergraph with bounded degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. showed in 1983. While Cooley et al. showed it for the 2-color case recently and independently from the author, our theorem contains the multicolor case and our proof is simple and provides a stronger embedding le...

We prove that the maximum number of edges in a k-uniform hypergraph on n vertices containing no 2-regular subhypergraph is ( n−1 k−1 ) if k ≥ 4 is even and n is sufficiently large. Equality holds only if all edges contain a specific vertex v. For odd k we conjecture that this maximum is ( n−1 k−1 ) + bn−1 k c, with equality only for the hypergraph described above plus a maximum matching omittin...

A 3-uniform hypergraph is called a minimum 3-tree, if for any 3coloring of its vertex set there is a heterochromatic triple and the hypergraph has the minimum possible number of triples. There is a conjecture that the number of triples in such 3-tree is dn(n−2) 3 e for any number of vertices n. Here we give a proof of this conjecture for any n ≡ 0, 1mod 12.

There are two possible definitions of the “s-disjoint r-uniform Kneser hypergraph” of a set system T : The hyperedges are either r-sets or r-multisets. We point out that Ziegler’s (combinatorial) lower bound on the chromatic number of an s-disjoint r-uniform Kneser hypergraph only holds if we consider r-multisets as hyperedges. We give a new proof of his result and show by example that a simila...