Georgia Institute of Technology College of Sciences

The original notion of poset dimension is due to Dushnik and Miller (1941). Last year, Uerckerdt (2016) proposed a variant, called local dimension, which has garnered considerable interest. A local realizer of a poset P is a collection of partial linear extensions of P that cover the comparabilities and incomparabilities of P. The local dimension of P is the minimum frequency of a local realizer where frequency is the maximum multiplicity of an element of P. Hiraguchi (1955) proved that any poset with n points has dimension at most n/2, which is sharp. We prove that the local dimension of a poset with n points is O(n/log n). To show that this bound is best possible, we use probabilistic methods to prove the following stronger result which extends a theorem of Chung, Erdős, and Spencer (1983): There is an n-vertex bipartite graph in which each difference graph cover of the edges will cover one of the vertices Θ(n/log n) times. (This is joint work with Jinha Kim, Ryan R. Martin, Tomáš Masařı́k, Warren Shull, Andrew Uzzell, and Zhiyu Wang)

No Combinatorics Seminar, but many others of interest: (a) on Friday [September 29th, 1pm-2pm in Skiles 005] He Guo, will give an ACO Student Seminar on "Packing nearly optimal Ramsey R(3,t) Graphs" (b) on Thursday [September 28th, 11am-12am in Skiles 006] Tom Bohman will give an ACO colloquim talk on "Randomized Controlled Trials for Combinatorial Construction" (c) on Saturday and Sunday [September 30th and October 1st] Atlanta Lecture Series in Combinatorics and Graph Theory XX takes place at Georgia Tech, with featured speaker Paul Seymour

The original concept ofdimension for posets was formulatedby Dushnik and Miller in 1941 and hasbeen studied extensively in the literature.Over the years, a number of variant formsof dimension have been proposed withvarying degrees of interest and application.However, in the recent past, two variantshave received extensive attention. Theyare Boolean dimension and local dimension.This is the first of two talks on these twoconcepts, with the second talk givenby Heather Smith. In this talk, wewill introduce the two parameters and providemotivation for their study. We will alsogive some concrete examples andprove some basic inequalities.This is joint work with a GeorgiaTech team in which my colleaguesare Fidel Barrera-Cruz, Tom Prag,Heather Smith and Libby Taylor.

This talk will focus on tree automata, which are tools to analyze existential monadic second order properties of rooted trees. A tree automaton A consists of a finite set \Sigma of colours, and a map \Gamma: \mathbb{N}^\Sigma \rightarrow \Sigma. Given a rooted tree T and a colouring \omega: V(T) \rightarrow \Sigma, we call \omega compatible with automaton A if for every v \in V(T), we have \omega(v) = \Gamma(\vec{n}), where \vec{n} = (n_\sigma: \sigma \in \Sigma) and n_\sigma is the number of children of v with colour \sigma. Under the Galton-Watson branching process set-up, if p_\sigma denotes the probability that a node is coloured \sigma, then \vec{p} = (p_\sigma: \sigma \in \Sigma) is obtained as a fixed point of a system of equations. But this system need not have a unique fixed point. Our question attempts to answer whether a fixed point of such a system simply arises out of analytic reasons, or if it admits of a probabilistic interpretation. I shall formally defined interpretation, and provide a nearly complete description of necessary and sufficient conditions for a fixed point to not admit an interpretation, in which case it is called rogue.Joint work with Tobias Johnson and Fiona Skerman.