Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Turán numbers of $k$-dimensional simplicial complexes: for each $k \in \mathbb N$, there is a $λ_k \ge k^{-2k^2}$ such that for any $k$-complex $\mathcal{S}$, every $k$-complex on $n \ge n_0(\mathcal{S})$ vertices with at least $n^{k+1 - λ_k}$ facets contains a homeomorphic copy of $\mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $λ_1$ is a result of Mader from 1967, and the existence of $λ_2$ was suggested by Linial in 2006 and recently proved by Keevash-Long-Narayanan-Scott. We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, \dots, V_r$ is said to be $d$-trace-bounded if for each $2 \le i \le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 \cup V_2 \cup \dots \cup V_i$. Our second main result is the following estimate for the Turán numbers of degenerate trace-bounded hypergraphs: for all $r \ge 2$ and $d\in\mathbb N$, there is an $α_{r,d} \ge (5rd)^{1-r}$ such that for any $d$-trace-bounded $r$-partite $r$-graph $H$, every $r$-graph on $n \ge n_0(H)$ vertices with at least $n^{r - α_{r,d}}$ edges contains a copy of $H$. This strengthens a result of Conlon-Fox-Sudakov from 2009 who showed that such a bound holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).
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.
Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. In this paper, we study the embedded homology as well as the homology of the (lower-)associated simplicial complexes for hypergraphs. We generalize the discrete Morse functions on simplicial complexes. We study the discrete Morse functions on hypergraphs as well as the discrete Morse functions on the (lower-)associated simplicial complexes of the hypergraphs.
We formulate and prove inverse mixing lemmas in the settings of simplicial complexes and k-uniform hypergraphs. In the hypergraph setting, we extend results of Bilu and Linial for graphs. In the simplicial complex setting, our results answer a question of Parzanchevski et al.
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.
Weighted hypergraphs are generalizations of weighted simplicial complexes. In recent years, weighted Laplacians of weighted simplicial complexes have been studied. In 2016, as a generalization of the homology of simplicial complexes, the embedded homology of hypergraphs was constructed. In this paper, we generalize the weighted Laplacians of weighted simplicial complexes to weighted hypergraphs. We study the relations between the weighted Laplacians and the weighted embedded homology of weighted hypergraphs. We generalize the Hodge decompositions of weighted simplicial complexes to weighted hy…
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.
We prove that any quasirandom uniform hypergraph $H$ can be approximately decomposed into any collection of bounded degree hypergraphs with almost as many edges. In fact, our results also apply to multipartite hypergraphs and even to the sparse setting when the density of $H$ quickly tends to $0$ in terms of the number of vertices of $H$. Our results answer and address questions of Kim, Kühn, Osthus and Tyomkyn; and Glock, Kühn and Osthus as well as Keevash. The provided approximate decompositions exhibit strong quasirandom properties which is very useful for forthcoming applications. Our resu…
Given an arbitrary hypergraph $\mathcal{H}$, we may glue to $\mathcal{H}$ a family of hypergraphs to get a new hypergraph $\mathcal{H}'$ having $\mathcal{H}$ as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph $\mathcal{H}'$ is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying ni…
We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with fundamental cuts and cycles playing their usual roles.
A simplicial cell ball is a simplicial poset whose geometric realization is homeomorphic to a ball. Recently, Samuel Kolins gave a series of necessary conditions and sufficient conditions on $h$-vectors of simplicial cell balls, and characterized them up to dimension 6. In this paper, we extend Kolins' results. We characterize all possible $h$-vectors of simplicial cell balls in arbitrary dimension.
Let $\mathcal{H}$ be a hypergraph on a finite set $V$. A {\em cover} of $\mathcal{H}$ is a set of vertices that meets all edges of $\mathcal{H}$. If $W$ is not a cover of $\mathcal{H}$, then $W$ is said to be a {\em noncover} of $\mathcal{H}$. The {\em noncover complex} of $\mathcal{H}$ is the abstract simplicial complex whose faces are the noncovers of $\mathcal{H}$. In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray numbers. The bound is in terms of hypergraph domination numbers. Also, our proof idea is …
For a $k$-uniform hypergraph $\mathcal{H}$ on vertex set $\{1, ..., n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erdős--Rényi random $k$-uniform hypergraph, $\text{coker}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = ω(1/n^{k - 1})$, $\text{coker}(M(\mathcal{H}))$ is torsion-free.
We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of $k+1$ vertices forms an edge with some probability $p_k$ independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408-417]. We consider a higher-dimensional notion of connectedness on this new model according to …
In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given in \cite{VT}, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. In \cite{F1}, Fröberg shows that the chordal graphs corresponds to graph algebras, $R/I(\mc{G})$, with linear resolutions. We extend Fröberg's method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The defini…
Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d=2), and strictly contains the class of `strongly stable' hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decomposit…
In this paper, we study the notion of chordality and cycles in hypergraphs from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. However, there is no unified definition for cycle or chordality in hypergraphs in the literature, so we consider several generalizations of these notions and study their algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of hypergraphs. Also we show that if $\mathcal{C}$ is a hypergraph such that $\l…
We consider $k$-dimensional random simplicial complexes that are generated from the binomial random $(k+1)$-uniform hypergraph by taking the downward-closure, where $k\geq 2$. For each $1\leq j \leq k-1$, we determine when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension one up to $j$ vanish and the zero-th cohomology group is isomorphic to $\mathbb{F}_2$. This property is not deterministically monotone for this model of random complexes, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of the…
We show that if the moment-angle complex $\mathcal{Z}_K$ associated to a simplicial complex $K$ is homotopy equivalent to a connected sum of sphere products with two spheres in each product, then $K$ decomposes as the simplicial join of an $n$-simplex $Δ^n$ and a minimally non-Golod complex. In particular, we prove that $K$ is minimally non-Golod for every moment-angle complex $\mathcal{Z}_K$ homeomorphic to a connected sum of two-fold products of spheres, answering a question of Grbić, Panov, Theriault and Wu.
Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs
University of San Francisco（旧金山大学）；University of South Florida（南佛罗里达大学）；The University of Manchester（曼彻斯特大学）；IBM Corporation（国际商业机器公司（IBM））；Imperial College London（伦敦帝国学院）；RWTH Aachen University（亚琛工业大学）
