相关论文: Interpreting Random Hypergraphs in Pseudofinite Fi…

We study several extensions of the notion of perfect graphs to $k$-uniform hypergraphs.

Let V denote a set of N vertices. To construct a "hypergraph process", create a new hyperedge at each event time of a Poisson process; the cardinality K of this hyperedge is random, with arbitrary probability generating function r(x),…

In this paper, the author introduces the concept and basic properties of finite (commutative) hyperfields. Also, the author shows that, up to isomorphism, there are exactly 2 hyperfields of order 2; 5 hyperfields of order 3; 7 hyperfields…

Let $\Gamma$ be a finite graph and let $\Gamma^{\mathrm{e}}$ be its extension graph. We inductively define a sequence $\{\Gamma_i\}$ of finite induced subgraphs of $\Gamma^{\mathrm{e}}$ through successive applications of an operation called…

We construct a model in which the splitting number is large and every ultrafilter has a small subset with no pseudo-intersection.

We define standardized constructions of finite fields, and standardized generators of (multiplicative) cyclic subgroups in these fields. The motivation is to provide a substitute for Conway polynomials which can be used by various software…

We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…

In recent years hypergraphs have emerged as a powerful tool to study systems with multi-body interactions which cannot be trivially reduced to pairs. While highly structured methods to generate synthetic data have proved fundamental for the…

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…

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…

We present a method for learning multiple scene representations given a small labeled set, by exploiting the relationships between such representations in the form of a multi-task hypergraph. We also show how we can use the hypergraph to…

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. They have algorithmic implications in various areas of computer science, as well as…

In this paper we study the theories of the infinite-branching tree and the $r$-regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be realized by infinite ultraproducts of polynomial exact classes of…

Here we introduce simple structures for the analysis of complex hypergraphs, hypergraph animals. These structures are designed to describe the local node neighbourhoods of nodes in hypergraphs. We establish their relationships to lattice…

A random intersection graph is constructed by independently assigning a subset of a given set of objects $W,$ to each vertex of the vertex set $V$ of a simple graph $G.$ There is an edge between two vertices of $V,$ iff their respective…

Random intersection graphs have received much interest and been used in diverse applications. They are naturally induced in modeling secure sensor networks under random key predistribution schemes, as well as in modeling the topologies of…

The entropy of random graph ensembles has gained widespread attention in the field of graph theory and network science. We consider microcanonical ensembles of simple graphs with prescribed degree sequences. We demonstrate that the…

Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…

In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by…

We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some…