The notion of spectrum of first-order properties introduced by J. Spencer for Erdos-Renyi random graph is considered in relation to random uniform hypergraphs. We study properties of spectrum for first-order formulae with bounded quantifier depth and estimate the values of minimum and maximum points of this spectrum. We also consider the set of limit points of the spectrum and give a bound for its minimum value.
Let $\mathcal{H}$ be a connected $k$-uniform hypergraph on $n$ vertices and $m$ hyperedges. In [A.~Banerjee, On the spectrum of hypergraph, Linear Algebra and its Application, 614(2021), 82--110], Anirban Banerjee introduced a new adjacency matrix for hypergraphs. In this article we consider the corresponding signless Laplacian matrix $Q(\mathcal{H})$ and discuss about its spectrum.
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the (lower-)associated independence hypergraph of the random hypergraph in terms of the probability distributions of certain random simplicial complex and certain random independence hypergraph of Erdös-Rényi type. We construct a graded structure of the map algebra explicitly and give algorithms to generate random hypergraphs and random simplicial complexes.
In this paper, we give a reduced formula of the characteristic polynomial of $k$-uniform hypergraphs with a pendant edge. And the explicit characteristic polynomial and all distinct eigenvalues of $k$-uniform hyperpath are given.
We extend results of Gyárfás and Füredi on the largest monochromatic component in $r$-colored complete $k$-uniform hypergraphs to the setting of random hypergraphs. We also study long monochromatic loose cycles in $r$-colored random hypergraphs. In particular, we obtain a random analog of a result of Gyárfás, Sárközy, and Szemerédi on the longest monochromatic loose cycle in $2$-colored complete $k$-uniform hypergraphs.
We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. We obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergraph with Omega(n^{k-epsilon}) edges, adding omega(n^{k epsilon/2}) random edges makes the hypergraph almost surely non-2-colorable. This is essentially tight, since there is a 2-colorable hypergraph with Omega(n^{k-ε}) edges which almost surely remains 2-colorable even after adding o(n^{k ε/ 2}) random edges.
We study the isomorphism problem for random hypergraphs. We show that it is solvable in polynomial time for the binomial random $k$-uniform hypergraph $H_{n,p;k}$, for a wide range of $p$. We also show that it is solvable w.h.p. for random $r$-regular, $k$-uniform hypergraphs $H_{n,r;k},r=O(1)$.
We estimate the likely values of the chromatic and independence numbers of the random $r$-uniform $d$-regular hypergraph on $n$ vertices for fixed $r$, large fixed $d$, and $n \rightarrow \infty$.
Despite of the extreme success of the spectral graph theory, there are relatively few papers applying spectral analysis to hypergraphs. Chung first introduced Laplacians for regular hypergraphs and showed some useful applications. Other researchers treated hypergraphs as weighted graphs and then studied the Laplacians of the corresponding weighted graphs. In this paper, we aim to unify these very different versions of Laplacians for hypergraphs. We introduce a set of Laplacians for hypergraphs through studying high-ordered random walks on hypergraphs. We prove the eigenvalues of these Laplacia…
We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have different densities. In the graph case, we prove a stronger result, on packing a random graph with a fixed graph.
Answering in a strong form a question posed by Bollobás and Scott, in this paper we determine the discrepancy between two random k-uniform hypergraphs, up to a constant factor depending solely on k.
Let $S=\{n_1,n_2,...,n_t\}$ be a finite set of positive integers with $\min(S)\geq 3$ and $t\geq 2$. For any positive integers $s_1,s_2,...,s_t$, we construct a family of 3-uniform bi-hypergraphs ${\cal H}$ with the feasible set $S$ and $r_{n_i}=s_i, i=1,2,...,t$, where each $r_{n_i}$ is the $n_i$th component of the chromatic spectrum of ${\cal H}$. As a result, we solve one open problem for 3-uniform bi-hypergraphs proposed by Bujtás and Tuza in 2008. Moreover, we find a family of sub-hypergraphs with the same feasible set and the same chromatic spectrum as it's own. In particular, we obtain …
In this paper, we present an equitable partition theorem of tensors, which gives the relations between $H$-eigenvalues of a tensor and its quotient equitable tensor and extends the equitable partitions of graphs to hypergraphs. Furthermore, with the aid of it, some properties and $H$-eigenvalues of the generalized power hypergraphs are obtained, which extends some known results, including some results of Yuan, Qi and Shao.
We consider the following definition of connectivity in $k$-uniform hypergraphs: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We determine the threshold at which the random $k$-uniform hypergraph with edge probability $p$ becomes $j$-connected with high probability. We also deduce a hitting time result for the random hypergraph process -- the hypergraph becomes $j$-connected at exactly the moment when the last isolated $j$-set disappears. This generalises well-known results for graphs.
Let $F$ be a strictly $k$-balanced $k$-uniform hypergraph with $e(F)\geq |F|-k+1$ and maximum co-degree at least two. The random greedy $F$-free process constructs a maximal $F$-free hypergraph as follows. Consider a random ordering of the hyperedges of the complete $k$-uniform hypergraph $K_n^k$ on $n$ vertices. Start with the empty hypergraph on $n$ vertices. Successively consider the hyperedges $e$ of $K_n^k$ in the given ordering, and add $e$ to the existing hypergraph provided that $e$ does not create a copy of $F$. We show that asymptotically almost surely this process terminates at a hy…
For a hypergraph $H$, define its intersection spectrum $I(H)$ as the set of all intersection sizes $|E\cap F|$ of distinct edges $E,F\in E(H)$. In their seminal paper from 1973 which introduced the local lemma, Erdős and Lovász asked: how large must the intersection spectrum of a $k$-uniform $3$-chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with $k$. Despite the problem being reiterated several times over the years by Erdős and other researchers, …
Let $H$ be a random $k$-uniform $n$-vertex hypergraph where every $k$-tuple belongs to $H$ independently with probability $p$. We show that for some $\varepsilon_k > 0$, if $p \geq n^{-\varepsilon_k}$, then asymptotically almost surely $H$ contains an $\left( n , k , k - 1 \right)$-Steiner System. Our main tool is Keevash's method of Randomized Algebraic Constructions.
In this paper we show that $e/n$ is the sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs, for all $k\ge 4$. When $k=3$ we show that $1/n$ is an asymptotic threshold. We also determine thresholds for the existence of other types of Hamilton cycles.
We present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all-ones hypermatrix. Several of the ingredients along a possible path to this conjecture are established, and may be of independent interest in spectral hypergraph/hypermatrix theory. In particular, we provide a bound on the spectral radius of the symmetric Bernoulli hyperensemble, and show that the spectrum of the complete \(k\)-uniform hyper…
We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) we have $|E_{i-1}-E_i|=\ell$. We prove that for $\ell \le k\le 2\ell$, with high probability almost all edges of a random $k$-uniform hypergraph $H(n,p,k)$ with $p(n)\gg \log^2 n/n$ can be decomposed into edge disjoint type $\ell$ Hamilton cycles. We also provide sufficient condi…
In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Spanning subgraphs of random graphs, Combinatorics, Probability & Computing 9 (2000), no. 2, 125-148] can be adapted from random graphs to random $r$-uniform hypergaphs and provide sufficient conditions when a random $r$-uniform hypergraph $\mathcal{H}^{(r)}(n,p)$ contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube-hypergraphs, lattices, spheres and Hamilton cycles in hypergraphs. Moreover, we study univer…
Let a_1,...,a_k satisfy a_1+...+a_k=1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting A_1,...,A_k is the number one would expect to find in a random k-uniform hypergraph. Can we then infer that H is quasi-random? We show that the answer is negative if and only if a_1=...=a_k=1/k. This resolves an open problem raised in 1991 by Chung and Graham [J. AMS '91]. While hypergraphs satisfying the property corresponding to a_1=...=a_k=1/k are not necess…
It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric $r$-matrix $A$ has a symmetric spectrum if and only if $r$ is even and $A$ is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs w…
The phase transition in the size of the giant component in random graphs is one of the most well-studied phenomena in random graph theory. For hypergraphs, there are many possible generalisations of the notion of a component, and for all but the simplest example, the phase transition phenomenon was first proved by Cooley, Kang and Person. In this paper we build on this and determine the asymptotic size of the unique giant component.
We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as \emph{online sprinkling}. As an illustrative application of this method, we show that for any fixed integer $k\geq 3$, the binomial $k$-uniform random hypergraph $H^{k}_{n,p}$ contains $N:=(1-o(1))\binom{n-1}{k-1}p$ edge-disjoint perfect matchings, provided $p\geq \frac{\log^{C}n}{n^{k-1}}$, where $C:=C(k)$ is an integer depending only on $k$. Our result for $N$ is asymptotically best optimal and for $p$ is optimal up to the $polylog(n)$ factor.
In the random hypergraph $H_{n,p;k}$ each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle in which every pair of adjacent edges intersects in a single vertex. We prove that if $p n^{k-1}/\log n$ tends to infinity with $n$ then $$\lim_{\substack{n\to \infty 2(k-1) |n}}\Pr(H_{n,p;k}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$ This is asymptotically best possible.
In the random hypergraph H=H(n,p;3) each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges {x_i,y_i,x_{i+1}\} for i=1,2,...,n/2. We prove that there exists an absolute constant K>0 such that if p>K\log n/n^2 then lim_{n->oo 4 |n}}Pr(H(n,p;3) contains a loose Hamilton cycle)=1.
We prove a lower bound on the length of the longest $j$-tight cycle in a $k$-uniform binomial random hypergraph for any $2 \le j \le k-1$. We first prove the existence of a $j$-tight path of the required length. The standard "sprinkling" argument is not enough to show that this path can be closed to a $j$-tight cycle -- we therefore show that the path has many extensions, which is sufficient to allow the sprinkling to close the cycle.
We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edg…
Let $H_{n,(p_m)_{m=2,\ldots,M}}$ be a random non-uniform hypergraph of dimension $M$ on $2n$ vertices, where the vertices are split into two disjoint sets of size $n$, and colored by two distinct colors. Each non-monochromatic edge of size $m=2,\ldots,M$ is independently added with probability $p_m$. We show that if $p_2,\ldots,p_M$ are such that the expected number of edges in the hypergraph is at least $dn\ln n$, for some $d>0$ sufficiently large, then with probability $(1-o(1))$, one can find a proper 2-coloring of $H_{n,(p_m)_{m=2,\ldots,M}}$ in polynomial time. We present a polynomial tim…
This paper studies the propagation connectivity of a random hypergraph $\mathbb{G}$ containing both 2-edges and 3-hyperedges. We find an exact threshold of the propagation connectivity of $\mathbb{G}$: If $I_{ε,r}<-1$, then $\mathbb{G}$ is not propagation connected with high probability; while if $I_{ε,r}>-1$, then $\mathbb{G}$ is propagation connected with high probability, where $I_{ε,r}$ is a constant dependent on the parameters of 2 and 3-edge probabilities.
We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its zero-th and first homology groups with coefficients in $\mathbb{F}_2$ vanish. Although this is not intrinsically a monotone property, we show that it nevertheless has a single sharp threshold, and indeed prove a hitting time result relating the connectedness to the disappearance of the last minimal obstruction.
In this paper we focus on the problem of finding (small) subhypergraphs in a (large) hypergraph. We use this problem to illustrate that reducing hypergraph problems to graph problems by working with the 2-section is not always a reasonable approach. We begin by defining a generalization of the binomial random graph model to hypergraphs and formalizing several definitions of subhypergraph. The bulk of the paper focusses on determining the expected existence of these types of subhypergraph in random hypergraphs. We also touch on the problem of determining whether a given subgraph appearing in th…
We consider vertex colourings of $r$-uniform hypergraphs $H$ in the classical sense, that is such that no edge has all its vertices given the same colour, and $(2,2)$-colourings of $H$ in which the vertices in any edge are given exactly two colours. This is a special case of constrained colourings introduced by Bujtas and Tuza which, in turn, is a generalisation of Voloshin's colourings of mixed hypergraphs. We study, $χ(H)$, the classical chromatic number, and the $(2,2)$-spectrum of $H$, that is, the set of integers $k$ for which $H$ has a $(2,2)$-colouring using exactly $k$ colours. We pres…
Compared to the classical binomial random (hyper)graph model, the study of random regular hypergraphs is made more challenging due to correlations between the occurrence of different edges. We develop an edge-switching technique for hypergraphs which allows us to show that these correlations are limited for a large range of densities. This extends some previous results of Kim, Sudakov and Vu for graphs. From our results we deduce several corollaries on subgraph counts in random $d$-regular hypergraphs. We also prove a conjecture of Dudek, Frieze, Ruciński and Šileikis on the threshold for the …
We consider the following game, played on a $k$-uniform hypergraph $H$. There are $q$ colors available and two players take it in turns to color vertices. A partial coloring is proper if no edge is mono-chromatic. One player, A, wishes to color all the vertices and the other player, B, wishes to prevent this. The {\em game chromatic number} $χ_g(H)$ is the minimum number of colors for which A has a winning strategy. We consider this in the context of a random $k$-uniform hypergraph and prove upper and lower bounds that hold w.h.p.
Recently, Kupavskii~[{\it On random subgraphs of {K}neser and {S}chrijver graphs. J. Combin. Theory Ser. A, {\rm 2016}.}] investigated the chromatic number of random Kneser graphs $\KG_{n,k}(ρ)$ and proved that, in many cases, the chromatic numbers of the random Kneser graph $\KG_{n,k}(ρ)$ and the Kneser graph $\KG_{n,k}$ are almost surely closed. He also marked the studying of the chromatic number of random Kneser hypergraphs $\KG^r_{n,k}(ρ)$ as a very interesting problem. With the help of $\Z_p$-Tucker lemma, a combinatorial generalization of the Borsuk-Ulam theorem, we generalize Kupavskii'…
We give an algorithmic proof for the existence of tight Hamilton cycles in a random r-uniform hypergraph with edge probability p=n^{-1+eps} for every eps>0. This partly answers a question of Dudek and Frieze [Random Structures Algorithms], who used a second moment method to show that tight Hamilton cycles exist even for p=omega(n)/n (r>2) where omega(n) tends to infinity arbitrary slowly, and for p=(e+o(1))/n (r>3). The method we develop for proving our result applies to related problems as well.
In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. For $r\geq 3$, we prove an optimal stopping-time result that if edges are sequently added to an initially empty $r$-graph, then as soon as the minimum degree is at least 2, the hypergraph almost surely has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős--Rényi random $r$-graph, and we also show that the $2$-out random $r$-graph almost surely has such a cycle. We obtain similar results for \textit{weak Berge} cycles as well, thus resolvi…
Given integers $k,j$ with $1\le j \le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the `Pathfinder' algorithm, a depth-first search algorithm which discovers $j$-tight paths in a $k$-uniform hypergraph. We prove that, in the supercritical case, w…
We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,α,ε)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}αn^{k}$ we have $$e(A_1,\dots, A_k)=(1\pmε)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $α=α(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big…
We establish an inclusion relation between two uniform models of random $k$-graphs (for constant $k \ge 2$) on $n$ labeled vertices: $\mathbb G^{(k)}(n,m)$, the random $k$-graph with $m$ edges, and $\mathbb R^{(k)}(n,d)$, the random $d$-regular $k$-graph. We show that if $n\log n\ll m\ll n^k$ we can choose $d = d(n) \sim {km}/n$ and couple $\mathbb G^{(k)}(n,m)$ and $\mathbb R^{(k)}(n,d)$ so that the latter contains the former with probability tending to one as $n\to\infty$. This extends an earlier result of Kim and Vu about "sandwiching random graphs". In view of known threshold theorems on t…
Let $c$ be a positive constant. Suppose that $r=o(n^{5/12})$ and the members of $\binom{[n]}{r}$ are chosen sequentially at random to form an intersecting hypergraph $\mathcal{H}$. We show that whp $\mathcal{H}$ consists of a simple hypergraph $\mathcal{S}$ of size $Θ(r/n^{1/3})$, a distinguished vertex $v$ and all $r$-sets which contain $v$ and meet every edge of $\mathcal{S}$. This is a continuation of the study of such random intersecting systems started in [Electron. J. Combin, (2003) R29] where the case $r=O(n^{1/3})$ was considered. To obtain the stated result we continue to investigate …
In this paper we consider $j$-tuple-connected components in random $k$-uniform hypergraphs (the $j$-tuple-connectedness relation can be defined by letting two $j$-sets be connected if they lie in a common edge and consider the transitive closure; the case $j=1$ corresponds to the common notion of vertex-connectedness). We determine that the existence of a $j$-tuple-connected component containing $Θ(n^j)$ $j$-sets in random $k$-uniform hypergraphs undergoes a phase transition and show that the threshold occurs at edge probability $\tfrac{(k-j)!}{\binom{k}{j}-1}n^{j-k}$. Our proof extends the re…
Inspired by the study of loose cycles in hypergraphs, we define the \emph{loose core} in hypergraphs as a structure which mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergraph $H^r(n,p)$, the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes. Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analys…
An {\em $\ell$-offset Hamilton cycle} $C$ in a $k$-uniform hypergraph $H$ on~$n$ vertices is a collection of edges of $H$ such that for some cyclic order of $[n]$ every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) satisfies $|E_{i-1}\cap E_i|=\ell$ and every pair of consecutive edges $E_{i},E_{i+1}$ in $C$ satisfies $|E_{i}\cap E_{i+1}|=k-\ell$. We show that in general $\sqrt{e^{k}\ell!(k-\ell)!/n^k}$ is the sharp threshold for the existence of the $\ell$-offset Hamilton cycle in the random $k$-uniform hypergraph $H_{n,p}^{(k)}$. We also examine this st…
This is an extended version of the thesis presented to the Programa de Pós-Graduação em Matemática of the Departamento de Matemática, PUC-Rio, in September 2013, incorporating some suggestions from the examining commission. Random graphs (and more generally hypergraphs) have been extensively studied, including their first order logic. In this work we focus on certain specific aspects of this vast theory. We consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p=p(n)$. We …
We look at colourings of $r$-uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge $E$ in an $r$-uniform hypergraph $H$ whose vertices are coloured is the partition of $r$ induced by the colour classes of the vertices in $E$. Let $Q$ be a set of partitions of $r$. A $Q$-colouring of $H$ is a colouring of its vertices such that only patterns appearing in $Q$ are allowed. We first show that many known hypergraph colouring problems, including Ramsey theory, can be stated in the language of $Q$-colourings. Then, using as our …
Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an \emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor problems in quasi-random $k$-graphs with minimum degree $Ω(n^{k-1})$. In particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs $H(n)$ with minimum degree $Ω(n^2)$ and minimum codegree $Ω(n)$ but with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite $3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense …
Given $r$-uniform hypergraphs $G$ and $H$ the Turán number $\rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erdős-Rényi random $r$-uniform hypergraph, and $H=C_{2\ell}^{(r)}$, the $r$-uniform linear cycle of length $2\ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine $\rm ex(G_{n,p}^{(r)}, C_{2\ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $\ell$ …
In this paper we prove an optimal co-degrees resilience property for the binomial $k$-uniform hypergraph model $H_{n,p}^k$ with respect to perfect matchings. That is, for a sufficiently large $n$ which is divisible by $k$, and $p\geq C_k\log_n/n$, we prove that with high probability every subgraph $H\subseteq H^k_{n,p}$ with minimum co-degree (meaning, the number of supersets every set of size $k-1$ is contained in) at least $(1/2+o(1))np$ contains a perfect matching.
Let $L$ be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of $L$. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some maps on the space of probability functions on sub-hypergraphs of $L$. We study the compositions of these maps as well as their actions on the space of probability functions.
In this note, we analyze two random greedy processes on sparse random graphs and hypergraphs with a given degree sequence. First we analyze the matching process, which builds a set of disjoint edges one edge at a time; then we analyze the independent process, which builds an independent set of vertices one vertex at a time. We use the differential equations method and apply a general theorem of Warnke. Our main contribution is to significantly reduce the associated systems of differential equations and simplify the expression for the final size of the matching or independent set.
Let $r \ge 2$ be a fixed constant and let $ {\mathcal H}$ be an $r$-uniform, $D$-regular hypergraph on $N$ vertices. Assume further that $ D \to \infty$ as $N \to \infty$ and that degrees of pairs of vertices in ${\mathcal H}$ are at most $L$ where $L \ = D/ (\log N)^{ω(1)}$. We consider the random greedy algorithm for forming a matching in $ \mathcal{H}$. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when…
