Strong Jumps and Lagrangians of Non-Uniform Hypergraphs

The hypergraph jump problem and the study of Lagrangians of uniform hypergraphs are two classical areas of study in the extremal graph theory. In this paper, we refine the concept of jumps to strong jumps and consider the analogous problems over non-uniform hypergraphs. Strong jumps have rich topological and algebraic structures. The non-strong-jump values are precisely the densities of the hereditary properties, which include the Tur\'an densities of families of hypergraphs as special cases. Our method uses a generalized Lagrangian for non-uniform hypergraphs. We also classify all strong jump values for $\{1,2\}$-hypergraphs.Comment: 19 page

New results on word-representable graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$ for each $x\neq y$. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202. Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 2011.], in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of $n$-vertex word-representable graphs is $2^{\frac{n^2}{3}+o(n^2)}$

A superadditivity and submultiplicativity property for cardinalities of sumsets

For finite sets of integers A1, . . . ,An we study the cardinality of the n-fold sumset A1 + · · · + An compared to those of (n − 1)-fold sumsets A1 + · · · + Ai−1 + Ai+1 + · · · + An. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets

Counting independent sets in hypergraphs

Let $G$ be a triangle-free graph with $n$ vertices and average degree $t$. We show that $G$ contains at least $$e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t (\frac{1}{2}\ln t-1)}$$ independent sets. This improves a recent result of the first and third authors \cite{countingind}. In particular, it implies that as $n \to \infty$, every triangle-free graph on $n$ vertices has at least $e^{(c_1-o(1)) \sqrt{n} \ln n}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 = 0.208138..$. Further, we show that for all $n$, there exists a triangle-free graph with $n$ vertices which has at most $e^{(c_2+o(1))\sqrt{n}\ln n}$ independent sets, where $c_2 = 1+\ln 2 = 1.693147..$. This disproves a conjecture from \cite{countingind}. Let $H$ be a $(k+1)$-uniform linear hypergraph with $n$ vertices and average degree $t$. We also show that there exists a constant $c_k$ such that the number of independent sets in $H$ is at least $$e^{c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}}.$$ This is tight apart from the constant $c_k$ and generalizes a result of Duke, Lefmann, and R\"odl \cite{uncrowdedrodl}, which guarantees the existence of an independent set of size $\Omega(\frac{n}{t^{1/k}} \ln^{1/k}t)$. Both of our lower bounds follow from a more general statement, which applies to hereditary properties of hypergraphs

Consistent random vertex-orderings of graphs

Given a hereditary graph property $\mathcal{P}$, consider distributions of random orderings of vertices of graphs $G\in\mathcal{P}$ that are preserved under isomorphisms and under taking induced subgraphs. We show that for many properties $\mathcal{P}$ the only such random orderings are uniform, and give some examples of non-uniform orderings when they exist

Shearer's inequality and Infimum Rule for Shannon entropy and topological entropy

We review subbadditivity properties of Shannon entropy, in particular, from the Shearer's inequality we derive the "infimum rule" for actions of amenable groups. We briefly discuss applicability of the "infimum formula" to actions of other groups. Then we pass to topological entropy of a cover. We prove Shearer's inequality for disjoint covers and give counterexamples otherwise. We also prove that, for actions of amenable groups, the supremum over all open covers of the "infimum fomula" gives correct value of topological entropy.Comment: 12 page