Related papers: On exchangeable random variables and the statistic…
We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and,…
Szemer\'edi's regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemer\'edi's theorem on arithmetic progressions . In this note we revisit this lemma from…
The hypergraph regularity lemma -- the extension of Szemer\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several…
The celebrated Erdos, Faber and Lovasz conjecture may be stated as follows: Any linear hypergraph on v points has chromatic index at most v. We will introduce the linear intersection number of a graph, and use this number to give an…
For any fixed simple graph $H=(V,E)$ and any fixed $u>0$, we establish the leading order of the exponential rate function for the probability that the number of copies of $H$ in the Erd\H{o}s--R\'enyi graph $G(n,p)$ exceeds its expectation…
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in…
Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model…
A graph $G$ is $H$-free if it does not contain an induced subgraph isomorphic to $H$. The study of the typical structure of $H$-free graphs was initiated by Erd\H{o}s, Kleitman and Rothschild, who have shown that almost all $C_3$-free…
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…
Hoffman's bound is a well-known eigenvalue bound on the chromatic number of a graph. By interpreting this bound as a parameter, we show multiple applications of colorings attaining the bound (Hoffman colorings) for several notions of graph…
The intricate relations between elements in natural and human-made systems sustain the complex processes that shape our world, forming multiscale networks of interactions. These networks can be represented as graphs composed of nodes…
We present a graph model for a background independent, relational approach to spacetime emergence. The general idea and the graph main features, detailed in [1], are discussed. This is a combinatorial (dynamical) metric graph, colored on…
In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H os--Ko--Rado theorem, the Hilton--Milner theorem, a theorem due to Frankl concerning the…
A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory provides a law of large numbers: If the group satisfies…
We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding…
Let $H$ be a triple system with maximum degree $d>1$ and let $r>10^7\sqrt{d}\log^{2}d$. Then $H$ has a proper vertex coloring with $r$ colors such that any two color classes differ in size by at most one. The bound on $r$ is sharp in order…
A sequence of random variables is exchangeable if its joint distribution is invariant under variable permutations. We introduce exchangeable variable models (EVMs) as a novel class of probabilistic models whose basic building blocks are…
We determine the asymptotic behavior of the maximum subgraph density of large random graphs with a prescribed degree sequence. The result applies in particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of Hajek [IEEE…
We establish a "diagonal" ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg's correspondence principle, prove that any "large" set in $K$ contains…
The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as "flag algebras", while the semantic, geometric one is often…