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Series: Combinatorics Seminar

We consider the question of coloring the first n integers with two
colors in such a way as to avoid copies of a given arithmetic configuration (such
as 3 term arithmetic progressions, or solutions to x+y=z+w). We know from
results of Van der Waerden and others that avoiding such configurations
completely is a hopeless task if n is sufficiently large, so instead we look at
the question of finding colorings with comparatively few monochromatic copies of
the configuration. At the very least, can we do significantly better than just
closing our eyes and coloring randomly?
I will discuss some partial answers, experimental results, and conjectured
answers to these questions for certain configurations based on joint work with
Steven Butler and Ron Graham.

Series: Combinatorics Seminar

The graph minor structure theorem of Robertson and Seymour gives anapproximate characterization of which graphs do not contain some fixedgraph H as a minor. The theorem has found numerous applications,including Robertson and Seymour's proof of the polynomial timealgorithm for the disjoint paths problem as well as the proof ofWagner's conjecture that graphs are well quasi-ordered under the minorrelation. Unfortunately, the proof of the structure theorem isextremely long and technical. We will discuss a new proof whichgreatly simplifies the argument and makes the result more widelyaccessible. This is joint work with Ken-ichi Kawarabayashi.

Series: Combinatorics Seminar

We extend the theory of infinite matroids recently developed by Bruhn et al to a well-known classical result in finite matroids while using the theory of connectivity for infinitematroids of Bruhn and Wollan. We prove that every infinite connected matroid M determines a graph-theoretic decomposition tree whose vertices correspond to minors of M that are3-connected, circuits, or cocircuits, and whose edges correspond to 2-separations of M. Tutte and many other authors proved such a decomposition for finite graphs; Cunningham andEdmonds proved this for finite matroids and showed that this decomposition is unique if circuits and cocircuits are also allowed. We do the same for infinite matroids. The knownproofs of these results, which use rank and induction arguments, do not extend to infinite matroids. Our proof avoids such arguments, thus giving a more first principles proof ofthe finite result. Furthermore, we overcome a number of complications arising from the infinite nature of the problem, ranging from the very existence of 2-sums to proving the treeis actually graph-theoretic.

Series: Combinatorics Seminar

Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. In this talk we first review the history of such problems. We will then focus on a conjecture of Bollobas and Thomason that the vertices of any r-uniform hypergraphs with m edges can be partitioned into r sets so that each set meets at least rm/(2r-1) edges. We will show that for r=3 and m large we can get an even better bound than what the conjecture suggests.

Series: Combinatorics Seminar

The matching sequence of a graph is the sequence whose $k$th term counts the number of matchings of size $k$. The independent set (or stable set) sequence does the same for independent sets. Except in very special cases, the terms of these sequences cannot be calculated explicitly, and one must be content to ask questions about global behavior. Examples of such questions include: is the sequence unimodal? is it log-concave? where are the roots of its generating function? For the matching sequence, these questions are answered fairly completely by a theorem of Heilmann and Lieb. For the independent set sequence, the situation is less clear. There are some positive results, one startling negative result, and a number of basic open questions. In this talk I will review the known results, and describe some recent progress on the questions.

Series: Combinatorics Seminar

**PLEASE NOTE SPECIAL TIME**

Let C(G) denote the set of lengths of cycles in a graph G. In this talk I shall present the recent proofs of two conjectures of P. Erdos on cycles in sparse graphs. In particular, we show that if G is a graph of average degree d containing no cycle of length less than g, then as d -> \infty then |C(G)| = \Omega(d^{\lfloor (g - 1)/2 \rfloor}). The proof is then adapted to give partial results on three further conjectures of Erdos on cycles in graphs with large chromatic number. Specifically, Erd\H{o}s conjectured that a triangle-free graph of chromatic number k contains cycles of at least k^{2 - o(1)} different lengths as k \rightarrow \infty. We define the {\em independence ratio} of a graph G by \iota(G) := \sup_{X \subset V(G)} \frac{|X|}{\alpha(X)}, where \alpha(X) is the independence number of the subgraph of G induced by X. We show that if G is a triangle free graph and \iota(G) \geq k, then |C(G)| = \Omega(k^2 \log k). This result is sharp in view of Kim's probabilistic construction of triangle-free graphs with small independence number. A number of salient open problems will be presented in conclusion. This work is in part joint with B. Sudakov. Abstract

Series: Combinatorics Seminar

A perfect matching in a $k$-uniform hypergraph $H=(V,E)$ on $n$ vertices is a set of$n/k$ disjoint edges of $H$, whilea fractional perfect matching in $H$ is a function $w:E --> [0,1]$ such that for each $v\in V$ we have $\sum_{e\ni v} w(e) = 1.$ Given $n \ge 3$ and $3\le k\le n$, let $m$ be the smallest integer suchthat whenever the minimum vertex degree in $H$ satisfies $\delta(H)\ge m$ then $H$ contains aperfect matching, and let $m^*$ be defined analogously with respect to fractional perfectmatchings. Clearly, $m^*\le m$.We prove that for large $n$, $m\sim m^*$, and suggest an approach to determine $m^*$, andconsequently $m$, utilizing the Farkas Lemma. This is a joint work with Vojta Rodl.

Series: Combinatorics Seminar

\ell-sequences are periodic binary sequences {a_i} that arise from Feedback with Carry Shift Registers and in many other ways. A decimation of {a_i} is a sequence of the form {a_{di}}. Goresky and Klapper conjectured that for any prime p>13 and any \ell-sequence based on p, every pair of allowable decimations of {a_i} is cyclically
distinct. If true this would yield large families of binary sequences with ideal arithmetic cross correlations. The conjecture is essentially equivalent to the statement that if p>13 then
the mapping x \to Ax^d on \mathbb Z/(p) with (d,p-1)=1, p \nmid A, permutes the even residues
only if it is the identity mapping. We will report on the progress towards resolving this conjecture, focussing on our joint work with Bourgain, Paulhus and Pinner.

Series: Combinatorics Seminar

In this talk we will report a short and transparent solution for the covering cost of white--grey trees which play a crucial role in the algorithm of Bergeron et al. to compute the rearrangement distance between two multi-chromosomal genomes in linear time (Theor. Comput. Sci., 410:5300-5316, 2009). In the process it introduces a new center notion for trees, which seems to be interesting on its own.

Series: Combinatorics Seminar

Let H be a fixed graph with h vertices. The graph removal lemma
states that every graph on n vertices with o(n^h) copies of H can be made
H-free by removing o(n^2) edges. We give a new proof which avoids
Szemeredi’s regularity lemma and gives a better bound. This approach also
works to give improved bounds for the directed and multicolored analogues
of the graph removal lemma. This answers questions of Alon and Gowers.