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Series: Combinatorics Seminar

We develop an information-theoretic foundation for compound Poisson
approximation and limit theorems (analogous to the corresponding
developments for the central limit theorem and for simple Poisson
approximation). First, sufficient conditions are given under which the
compound Poisson distribution has maximal entropy within a natural
class of probability measures on the nonnegative integers. In
particular, it is shown that a maximum entropy property is valid
if the measures under consideration are log-concave, but that it
fails in general. Second, approximation bounds in the (strong)
relative entropy sense are given for distributional approximation
of sums of independent nonnegative integer valued random variables
by compound Poisson distributions. The proof techniques involve the
use of a notion of local information quantities that generalize the
classical Fisher information used for normal approximation, as well
as the use of ingredients from Stein's method for compound Poisson
approximation. This work is joint with Andrew Barbour (Zurich),
Oliver Johnson (Bristol) and Ioannis Kontoyiannis (AUEB).

Series: Combinatorics Seminar

Let G be a graph and K be a field. We associate to G a projective toric variety X_G over K, the cut variety of the graph G. The cut ideal I_G of the graph G is the ideal defining the cut variety. In this talk, we show that, if G is a subgraph of a subdivision of a book or an outerplanar graph, then the minimal generators are quadrics. Furthermore we describe the generators of the cut ideal of a subdivision of a book.

Series: Combinatorics Seminar

The entropy function has a number of nice properties that make it a useful
counting tool, especially when one wants to bound a set with respect to the set's
projections. In this talk, I will show a method developed by Mokshay Madiman, Prasad
Tetali, and myself that builds on the work of Gyarmati, Matolcsi and Ruzsa as well as
the work of Ballister and Bollobas. The goal will be to give a black-box method for
generating projection bounds and to show some applications by giving new bounds on
the sizes of Abelian and non-Abelian sumsets.

Series: Combinatorics Seminar

I will present an approximation algorithm for the following problem: Given a graph G and a parameter k, find k edges to add to G as to maximize its algebraic connectivity. This problem is known to be NP-hard and prior to this work no algorithm was known with provable approximation guarantee. The algorithm uses a novel way of sparsifying (patching) part of a graph using few edges.

Series: Combinatorics Seminar

Choose a graph uniformly at random from all d-regular graphs on n vertices. We determine the chromatic number of the graph for about half of all values of d, asymptotically almost surely (a.a.s.) as n grows large. Letting k be the smallest integer satisfying d < 2(k-1)\log(k-1), we show that a random d-regular graph is k-colorable a.a.s. Together with previous results of Molloy and Reed, this establishes the chromatic number as a.a.s. k-1 or k. If furthermore d>(2k-3)\log(k-1) then the chromatic number is a.a.s. k. This is an improvement upon results recently announced by Achlioptas and Moore. The method used is "small subgraph conditioning'' of Robinson and Wormald, applied to count colorings where all color classes have the same size. It is the first rigorously proved result about the chromatic number of random graphs that was proved by small subgraph conditioning. This is joint work with Xavier Perez-Gimenez and Nick Wormald.

Series: Combinatorics Seminar

Tolerance graphs were introduced in 1982 by Golumbic and Monma as a generalization of the class of interval graphs. A graph G= (V, E) is an interval graph if each vertex v \in V can be assigned a real interval I_v so that xy \in E(G) iff I_x \cap I_y \neq \emptyset. Interval graphs are important both because they arise in a variety of applications and also because some well-known recognition problems that are NP-complete for general graphs can be solved in polynomial time when restricted to the class of interval graphs. In certain applications it can be useful to allow a representation of a graph using real intervals in which there can be some overlap between the intervals assigned to non-adjacent vertices. This motivates the following definition: a graph G= (V, E) is a tolerance graph if each vertex v \in V can be assigned a real interval I_v and a positive tolerance t_v \in {\bf R} so that xy \in E(G) iff |I_x \cap I_y| \ge \min\{t_x,t_y\}. These topics can also be studied from the perspective of ordered sets, giving rise to the classes of Interval Orders and Tolerance Orders. In this talk we give an overview of work done in tolerance graphs and orders . We use hierarchy diagrams and geometric arguments as unifying themes.

Series: Combinatorics Seminar

This is joint work with Alex Postnikov. Imagine that you are to build a system of space stations (graph vertices), which communicate via laser beams (edges). The edge directions were already chosen, but you must place the stations so that none of the beams miss their targets. In this talk, we let the edge directions be generic and independent, a choice that constrains vertex placement the most. For K_{3} in \mathbb{R}^{2}, the edges specify a unique triangle, but its size is arbitrary --- D_{2}(K_{3})=1 degree of freedom; we say that K_{3} is rigid in \mathbb{R}^{2}. We call D_{n}(G) the degree of parallel rigidity of the graph for generic edge directions. We found an elegant combinatorial characterization of D_{n}(G) --- it is equal to the minimal number of edges in the intersection of n spanning trees of G. In this talk, I will give a linear-algebraic proof of this result, and of some other properties of D_{n}(G). The notion of parallel graph rigidity was previously studied by Whiteley and Develin-Martin-Reiner. The papers worked with the generic parallel rigidity matroid; I will briefly compare our results in terms of D_{n}(G) with the previous work.

Series: Combinatorics Seminar

We show that for all \el an \epsilon>0 there is a constant c=c(\ell,\epsilon)>0 such that every \ell-coloring of the triples of an N-element set contains a subset S of size c\sqrt{\log N} such that at least 1-\epsilon fraction of the triples of S have the same color. This result is tight up to the constant c and answers an open question of Erd\H{o}s and Hajnal from 1989 on discrepancy in hypergraphs. For \ell \geq 4 colors, it is known that there is an \ell-coloring of the triples of an N-element set whose largest monochromatic subset has cardinality only \Theta(\log \log N). Thus, our result demonstrates that the maximum almost monochromatic subset that an \ell-coloring of the triples must contain is much larger than the corresponding monochromatic subset. This is in striking contrast with graphs, where these two quantities have the same order of magnitude. To prove our result, we obtain a new upper bound on the \ell-color Ramsey numbers of complete multipartite 3-uniform hypergraphs, which answers another open question of Erd\H{o}s and Hajnal. (This is joint work with D. Conlon and J. Fox.)

Series: Combinatorics Seminar

In this work (joint with Derrick Hart), we show that there exists a constant c > 0 such that the following holds for all n sufficiently large: if S is a set of n monic polynomials over C[x], and the product set S.S = {fg : f,g in S}; has size at most n^(1+c), then the sumset S+S = {f+g : f,g in S}; has size \Omega(n^2). There is a related result due to Mei-Chu Chang, which says that if S is a set of n complex numbers, and |S.S| < n^(1+c), then |S+S| > n^(2-f(c)), where f(c) -> 0 as c -> 0; but, there currently is no result (other than the one due to myself and Hart) giving a lower bound of the quality >> n^2 for |S+S| for a fixed value of c. Our proof combines combinatorial and algebraic methods.

Series: Combinatorics Seminar

The study of random tilings of planar lattice regions goes back to the solution of the dimer model in the 1960's by Kasteleyn, Temperley and Fisher, but received new impetus in the early 1990's, and has since branched out in several directions in the work of Cohn, Kenyon, Okounkov, Sheffield, and others. In this talk, we focus on the interaction of holes in random tilings, a subject inspired by Fisher and Stephenson's 1963 conjecture on the rotational invariance of the monomer-monomer correlation on the square lattice. In earlier work, we showed that the correlation of a finite number of holes on the triangular lattice is given asymptotically by a superposition principle closely paralleling the superposition principle for electrostatic energy. We now take this analogy one step further, by showing that the discrete field determined by considering at each unit triangle the average orientation of the lozenge covering it converges, in the scaling limit, to the electrostatic field. Our proof involves a variety of ingredients, including Laplace's method for the asymptotics of integrals, Newton's divided difference operator, and hypergeometric function identities.