We present a variant of a universality result of Rödl [On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), no. 1-2, 125-134] for sparse, $3$-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom'' uniform hypergraphs, which is proved in the companion paper [Counting results for sparse pseudorandom hypergraphs I].
Counting results for sparse pseudorandom hypergraphs I
我们建立了一个所谓的计数引理，它允许将某些线性均匀超图嵌入到稀疏伪随机超图中，推广了图的结果[Embedding graphs with bounded degrees in sparse pseudorandom graphs，Israel J.Math.139（2004），93-137]。我们的结果的应用在配套文件[稀疏伪随机超图计数结果II]。
A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is `pseudorandom', meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.
Amongst $d$-regular $r$-uniform hypergraphs on $n$ vertices, which ones have the largest number of independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what the correct general conjecture ought to be; our goal here is propose such a generalisation. Lending credence to our conjecture, we verify it within the class of `quasi-bipartite' hypergraphs (a generalisation of bipartite graphs that seems natural in this context) by adopting the entropic approach of Kahn.
The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to sparse but pseudorandom hypergraphs. We use this framework to give a proof of hypergraph removal for sub-hypergraphs of sparse random hypergraphs.
We study sparse hypergraphs which satisfy a mild pseudorandomness condition known as $L_p$ regularity. We prove appropriate regularity and counting lemmas, and we extend the relative removal lemma of Tao in this setting. This answers a question of Borgs, Chayes, Cohn and Zhao.
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 show that for all $0\le \ell<k-1$, every $k$-graph with minimum co-degree $δn$ with $δ>1/2$ has (asymptotically and up to a subexponential factor) at least as many Hamiltonian $\ell$-cycles as in a typical random $k$-graph with edge-probability $δ$. This significantly improves a recent result of Glock, Gould, Joos, Kühn, and Osthus, and verifies a conjectur…
A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. Rödl, Ruciński, and Szemerédi proved that for $k\geq 3$, every $k$-graph on $n$ vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such $k$-graphs is $\exp(n\ln n-Θ(n))$. As a corollary, we obtain a similar estimate on the number of Hamilton $\ell$-cycles in such $k$-graphs for all $\ell\in\{0,\dots,k-1\}$, which makes prog…
We consider some variants of the Gowers box norms, introduced by Hatami, and show their relevance in the context of sparse hypergraphs. Our main results are the following. Firstly, we prove a generalized von Neumann theorem for $L_p$ graphons. Secondly, we give natural examples of pseudorandom families, that is, sparse weighted uniform hypergraphs which satisfy relative versions of the counting and removal lemmas.
The Catalan numbers (C_n)_{n >= 0} = 1,1,2,5,14,42,... form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we define an infinite collection of generalizations C_n^(m)…
The paper deals with a problem of Additive Combinatorics. Let ${\mathbf G}$ be a finite abelian group of order $N$. We prove that the number of subset triples $A,B,C\subset {\mathbf G}$ such that for any $x\in A$, $y\in B$ and $z\in C$ one has $x+y\ne z$ equals $$ 3\cdot 4^N+N3^{N+1} + O((3-c_*)^N) $$ for some absolute constant $c_*>0$. This provides a tight estimate for the number of independent sets in a special 3-uniform linear hypergraph and gives a support for the natural conjecture concerning the maximal possible number of independent sets in such hypergraphs on $n$ vertices.
We give a fully polynomial-time approximation scheme (FPTAS) to count the number of $q$-colorings for $k$-uniform hypergraphs with maximum degree $Δ$ if $k\ge 28$ and $q >357Δ^{\frac{14}{k-14}}$ . We also obtain a polynomial-time almost uniform sampler if $q>931Δ^{\frac{16}{k-16/3}}$. These are the first approximate counting and sampling algorithms in the regime $q\llΔ$ (for large $Δ$ and $k$) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of $k$ and $Δ$ in Moitra's approach.
We consider the problem of counting $k$-cliques in $s$-uniform Erdos-Renyi hypergraphs $G(n,c,s)$ with edge density $c$, and show that its fine-grained average-case complexity can be based on its worst-case complexity. We prove the following: 1. Dense Erdos-Renyi graphs and hypergraphs: Counting $k$-cliques on $G(n,c,s)$ with $k$ and $c$ constant matches its worst-case time complexity up to a $\mathrm{polylog}(n)$ factor. Assuming randomized ETH, it takes $n^{Ω(k)}$ time to count $k$-cliques in $G(n,c,s)$ if $k$ and $c$ are constant. 2. Sparse Erdos-Renyi graphs and hypergraphs: When $c = Θ(n^…
In breakthrough results, Saxton-Thomason and Balogh-Morris-Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton-Thomason and an entropy-based framework to deduce container and counting theorems for hereditary properties of k-colourings of very general objects, which include both vertex- and edge-colourings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterisation and transference results for hereditary proper…
In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we prove a version of this result where we assume that $r$-tuples of vertices in $G$ have their neighbours evenly distributed in $G$. Third, we also prove a counting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$ satisfies our density bound, and present some open problems. A st…
An Analytic counting Framework for the generalised Erdős box problem
School of Mathematical Sciences National Institute of Science Education and Research (NISER)（国家科学教育研究所数学科学学院）；Homi Bhabha National Institute (HBNI) Training School Complex（霍米·巴巴国立学院培训学院）
Given complex numbers $w_1, \ldots, w_n$, we define the weight $w(X)$ of a set $X$ of 0-1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors $(x_1, \ldots, x_n)$ in $X$. We present an algorithm, which for a set $X$ defined by a system of homogeneous linear equations with at most $r$ variables per equation and at most $c$ equations per variable, computes $w(X)$ within relative error $ε>0$ in $(rc)^{O(\ln n-\ln ε)}$ time provided $|w_j| \leq β(r \sqrt{c})^{-1}$ for an absolute constant $β>0$ and all $j=1, \ldots, n$. A similar algorithm is constructed for computing the weight of…
