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Series: Combinatorics Seminar

A set partition of [n] can be represented graphically by drawing n dots on a
horizontal line and connecting the points in a same block by arcs. Crossings
and nestings are then pairs of arcs that cross or nest. Let G be an abelian
group, and \alpha, \beta \in G. In this talk I will look at the distribution
of the statistic s_{\alpha, \beta} = \alpha * cr + \beta * ne on subtrees of
the tree of all set partitions and present a result which says that the
distribution of s_{\alpha, \beta} on a subtree is determined by its
distribution on the first two levels.

Series: Combinatorics Seminar

Given a finite poset $P$, we consider the largest size ${\rm La}(n,P)$ of a family of subsets of $[n]:=\{1,\ldots,n\}$ that contains no subposet $HP. Sperner's Theorem (1928) gives that ${\rm La}(n,P_2)= {n\choose{\lfloor n/2\rfloor}}$, where $P_2$ is the two-element chain. This problem has been studied intensively in recent years, and it is conjectured that $\pi(P):= \lim_{n\rightarrow\infty} {\rm La}(n,P)/{n\choose{\lfloor n/2\rfloor}}$ exists for general posets $P$, and, moreover, it is an integer. For $k\ge2$ let $D_k$ denote the $k$-diamond poset $\{A< B_1,\ldots,B_k < C\}$. We study the average number of times a random full chain meets a $P$-free family, called the Lubell function, and use it for $P=D_k$ to determine $\pi(D_k)$ for infinitely many values $k$. A stubborn open problem is to show that $\pi(D_2)=2$; here we prove $\pi(D_2)<2.273$ (if it exists). This is joint work with Wei-Tian Li and Linyuan Lu of University of South Carolina.

Series: Combinatorics Seminar

Many of the simplest and easiest implemented approximation algorithms can be thought of as derandomizations of the naive random algorithm. Here we consider the question of whether performing the algorithm on a random reordering of the variables provides an improvement in the worst case expected performance. (1) For Johnson's algorithm for Maximum Satisfiability, we show this is indeed the case: While in the worst case Johnson's algorithm only provides a 2/3 approximation, the additional randomization step guarantees a 2/3+c approximation for some positive c. (2) For the greedy algorithm for MAX-CUT, we show to the contrary that the randomized version does NOT provide a 1/2+c approximation for any c on general graphs. This is in contrast to a result of Mathieu and Schudy showing it provides a 1-epsilon approximation on dense graphs. Joint with Asaf Shapira and Prasad Tetali.

Series: Combinatorics Seminar

In the paper "On the Size of Maximal Chains and the Number of Pariwise Disjoint Maximal Antichains" Duffus and Sands proved the following:If P is a poset whose maximal chain lengths lie in the interval [n,n+(n-2)/(k-2)] for some n>=k>=3 then there exist k disjoint maximal antichains in P. Furthermore this interval is tight. At the end of the paper they conjecture whether the dual statement is true (swap the words chain and antichain in the theorem). In this talk I will prove the dual and if time allows I will show a stronger version of both theorems.

Series: Combinatorics Seminar

In the talk we will consider the problem of deciding whether agiven $r$-uniform hypergraph $H$ with minimum vertex degree atleast $c|V(H)|$, has a vertex 2-coloring. This problem has beenknown also as the Property B of a hypergraph. Motivated by an oldresult of Edwards for graphs, we summarize what can be deducedfrom his method about the complexity of the problem for densehypergraphs. We obtain the optimal dichotomy results for2-colorings of $r$-uniform hypergraphs when $r=3,4,$ and 5. During the talk we will present the NP-completeness results followed bypolynomial time algorithms for the problems above the thresholdvalue. The coloring algorithms rely on the known Tur\'{a}n numbersfor graphs and hypergraphs and the hypergraph removal lemma.

Series: Combinatorics Seminar

Erd\H{o}s and R\'enyi observed that a curious phase transition in the size of the largest component in arandom graph G(n,p): If pn < 1, then all components have size O(\log n), while if pn > 1 there exists a uniquecomponent of size \Theta(n). Similar transitions can be seen to exist when taking random subgraphs of socalled (n,d,\lambda) graphs (Frieze, Krivelevich and Martin), dense graphs (Bollobas et. al) and several otherspecial classes of graphs. Here we consider the story for graphs which are sparser and irregular. In thisregime, the answer will depend on our definition of a 'giant component'; but we will show a phase transitionfor graphs satisfying a mild spectral condition. In particular, we present some results which supersede ourearlier results in that they have weaker hypotheses and (in some sense) prove stronger results. Additionally,we construct some examples showing the necessity of our new hypothesis.

Series: Combinatorics Seminar

The Faber-Krahn problem for the cube deals with understanding the function,
Lambda(t) = the maximal eigenvalue of an induced t-vertex subgraph of the
cube (maximum over all such subgraphs).
We will describe bounds on Lambda(t), discuss connections to isoperimetry
and coding theory, and present some conjectures.

Series: Combinatorics Seminar

In 1963 Fisher and Stephenson conjectured that the correlation function of two oppositely colored monomers in a sea of dimers on the square lattice is rotationally invariant in the scaling limit. More precisely, the conjecture states that if one of the monomers is fixed and the other recedes to infinity along a fixed ray, the correlation function is asymptotically $C d^(-1/2)$, where $d$ is the Euclidean distance between the monomers and $C$ is a constant independent of the slope of the ray. Shortly afterward Hartwig rigorously determined $C$ when the ray is in a diagonal direction, and this remains the only direction settled in the literature. We generalize Hartwig's result to any finite collection of monomers along a diagonal direction. This can be regarded as a counterpart of a result of Zuber and Itzykson on n-spin correlations in the Ising model. A special case proves that two same-color monomers interact the way physicists predicted.

Series: Combinatorics Seminar

I will divide the talk between two topics. The first is Stirling numbers of the second kind, $S(n,k)$. For each $n$ the maximum $S(n,k)$ is achieved either at a unique $k=K_n$, or is achieved twice consecutively at $k=K_n,K_n+1$. Call those $n$ of the second kind {\it exceptional}. Is $n=2$ the only exceptional integer? The second topic is $m\times n$ nonnegative integer matrices all of whose rows sum to $s$ and all of whose columns sum to $t$, $ms=nt$. We have an asymptotic formula for the number of these matrices, valid for various ranges of $(m,s;n,t)$. Although obtained by a lengthy calculation, the final formula is succinct and has an interesting probabilistic interpretation. The work presented here is collaborative with Carl Pomerance and Brendan McKay, respectively.

Series: Combinatorics Seminar

Let a_1,...,a_k satisfy a_1+...+a_k=1 and suppose a k-uniform hypergraph on n
vertices satisfies the following property; in any partition of its vertices into k
sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting
A_1,...,A_k is the number one would expect to find in a random k-uniform hypergraph.
Can we then infer that H is quasi-random? We show that the answer is negative if and
only if a_1=...=a_k=1/k. This resolves an open problem raised in 1991 by Chung and
Graham [J. AMS '91].
While hypergraphs satisfying the property corresponding to a_1=...=a_k=1/k are not
necessarily quasi-random, we manage to find a characterization of the hypergraphs
satisfying this property. Somewhat surprisingly, it turns out that (essentially)
there is a unique non quasi-random hypergraph satisfying this property. The proofs
combine probabilistic and algebraic arguments with results from the theory of
association schemes.
Joint work with Raphy Yuster