Related papers: Hypergraph limits: a regularity approach

We introduce a parameterized family of invariants for $\ell$-uniform hypergraphs. To each $\mathbb{K}$-linear transformation $T:\mathbb{K}^{\ell}\to \mathbb{K}^r$ we associate a function $\mathrm{Sig}(-,T)$ that maps $\ell$-uniform…

Let K be a field, [n]= {1,...,n} and H=([n],E) be a hypergraph. For an integer d >= 1 the Lovasz-Saks-Schrijver ideal (LSS-ideal) L_H^K (d) in K[y_{ij}~:~(i,j) \in [n] x [d]] is the ideal generated by the polynomials $f^{(d)}_{e}=…

We obtain a hypergraph generalisation of the graph blow-up lemma proved by Komlos, Sarkozy and Szemeredi, showing that hypergraphs with sufficient regularity and no atypical vertices behave as if they were complete for the purpose of…

Let $T(H, G_n)$ be the number of monochromatic copies of a fixed connected graph $H$ in a uniformly random coloring of the vertices of the graph $G_n$. In this paper we give a complete characterization of the limiting distribution of $T(H,…

For $0\leq \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph $H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments of length $k$ and every two consecutive edges overlap in exactly $\ell$ vertices. We…

Let $r,s,t\geq2$ be integers. For $r$-graphs $G$ and $F_1,\dots,F_s$, we write $G\to(F_1,\dots,F_s)$ if every $s$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$-th color for some $1\leq i\leq s$. Let…

A celebrated result of R\"odl and Ruci\'nski states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq…

A directed hypergraph is a hypergraph in which the vertex set of each hyperedge is partitioned into two disjoint parts, a head and a tail. Keszegh and P\'alv\"olgyi posed the following conjecture. Let $H$ be a directed hypergraph such that…

A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring such that no two vertices in the same edge have the same color, and each $k$-element subset of colors appears on at most one edge. The harmonious number $h(H)$ is…

Graph homomorphism has been an important research topic since its introduction [17]. Stated in the language of binary relational structures in that paper [17], Lov\'asz proved a fundamental theorem that, for a graph $H$ given by its $0$-$1$…

We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any $n$-vertex, $d$-regular graph $G$ and any graph $H$ (possibly with loops), \[\hom(G,H) \leq \max\left\lbrace\hom(K_{d,d}, H)^{\frac{n}{2d}},…

We study Ramsey properties of randomly perturbed $3$-uniform hypergraphs. For~$t\geq 2$, write $\tilde K^{(3)}_t$ to denote the $3$-uniform {\it expanded} clique hypergraph obtained from the complete graph $K_t$ by expanding each of the…

We consider the Erd{\H{o}}s - Faber - Lov\'{a}sz (EFL) conjecture for hypergraphs. This paper gives an upper bound for the chromatic number of $r$ regular linear hypergraphs $\textbf{H}$ of size $n$. If $r \ge 4$, $\chi (\textbf{H}) \le…

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…

Given graphs $G, H_1, H_2$, we write $G \rightarrow ({H}_1, H_2)$ if every $\{$red, blue$\}$-coloring of the edges of $G$ contains a red copy of $H_1$ or a blue copy of $H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G…

A copy of a hypergraph $F$ is called an $F$-copy. Let $K_k^r$ denote the complete $r$-uniform hypergraph whose vertex set is $[k] = \{1, \dots, k\}$ (that is, the edges of $K_k^r$ are the $r$-element subsets of $[k]$). Given an $r$-uniform…

Sidorenko's conjecture states that the number of copies of a bipartite graph $H$ in a graph $G$ is asymptotically minimised when $G$ is a quasirandom graph. A notorious example where this conjecture remains open is when $H=K_{5,5}\setminus…

We say that a k-uniform hypergraph C is an l-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of…

The classical sharp threshold theorem of Friedgut and Kalai (1996) asserts that any symmetric monotone function $f:\{0,1\}^{n}\to\{0,1\}$ exhibits a sharp threshold phenomenon. This means that the expectation of $f$ with respect to the…

We introduce the 2-sorted counting logic $GC^k$ that expresses properties of hypergraphs. This logic has available k variables to address hyperedges, an unbounded number of variables to address vertices, and atomic formulas E(e,v) to…