The algebraic theory of finite automata will be overviewed-- no prerequisites.

 

More recent computational models will be described. Recent results and

problems (some longstanding, most more recent) in the area will be described..

 

All are welcome.

Online Ramsey theory is the study of a family of combinatorial games between Builder and Painter. Starting with an infinite set of independent vertices, Builder presents an edge to Painter. Painter responds by coloring the edge red or blue. Builder continues to present edges, and Painter colors each edge as it is presented. Builder's goal is to obtain a monochromatic copy of some target graph G, and Painter tries to stop Builder. Classic Ramsey theory shows that if Builder is allowed to present large cliques, then Builder wins. Therefore, we restrict Builder by requiring that at all times, the presented graph belongs to a family of graphs H. This defines the game (G, H).

 

We examine such games in the case that H=S_k, where S_k is the class of graphs with maximum degree at most k. For each graph G, define the online degree- Ramsey number odr(G) to be the minimum k such that builder wins (G, S_k). We characterize the graphs G with odr(G) <= 3. If G is a cycle, then odr(G) <= 5. If in addition G is an even cycle, a triangle, or a sufficiently large odd cycle, then odr(G) <= 4. We conclude with some open problems.

for each value of n.

some preliminary results and plans for future study.  

Upper Bounds for the length of $s$- Extremal Codes over

$\mathbb F_2$, $\mathbb F_4$, and $\mathbb F_2 + u\mathbb F_2$

$s$- extremal codes.

I will present an optimal stopping time for directed path of order n.

It is known that in a finite semimodular lattice the join of

a profile of elements need not be an upper bound for the set of medians

for that profile.

 

We shall present some preliminary results that show if a semimodular

lattice is "not very tall," then the join of a given profile of elements

does provide an upper bound for the set of medians for that profile.

Definitions and examples on the topic above will be reviewed.

Recent results on asymptotic classes of free spectra will be presented,

along with a related counting problem for so-called generalized posets.

All are welcome. No prerequisites.

I will describe a classical algebraic invariant, the free spectum,

whereby to each finite algebra A is associated a sequence of

positive integers (f_n(A)). In the talk the free spectrum will be

defined and numerous examples given, including from basic algebraic

structures such as groups and rings.

Some of the techniques for determing the asymptotics of free spectra

have a combinatorics flavor: an open counting problem for a class of

structures that generalize posets will be presented.

There are no prerequisites for the talk.

I will present the motivation, background, and

evidence for a recent conjecture by Furedi, Gyarfas, and Simonyi

(Combinatorics, Probability, and Computing (2005) 14, 435-438).

A set of pairwise disjoint edges M in a graph G is a connected matching

if, for any pair of edges e and f in M, there exists an edge in G

incident to both e and f. Furedi et al. conjecture that, every graph with

4t-1 vertices and whose complement contains no triangle must contain

a connected matching with at least t edges. This bound, if true, would be

sharp as seen by considering the graph that is the union of two copies of the
complete graph on 2t-1 vertices.

A binary code is called formally self-dual (f.s.d.) if its weight enumerator

is the same as its dual code. Formally self-dual codes have been of interest since sometimes

they have a larger minimum distance than self-dual codes of the same length.

An f.s.d. code is called even if all weights of the codewords are even, and odd otherwise.

In this talk I will begin with basic definitions of f.s.d. codes

and describe the current known status of classification of extremal/maximal f.s.d. even codes.

I will also describe a conjecture claiming that there does not exist a near-extremal f.s.d. even code

whose length is at least 48 and divisible by 8. This conjecture has been proposed in our joint

paper with Professor Vera Pless.

Given a finite group G, for a positive integer n, the positive integer fnG is the number  of distinct functions from Gn → G which

can be represented by words in the alphabet {x1,...,xn}.  There are only |G||G|^|G| functions of n-variables from G to itself; so fnG <=  |G||G|^|G|.  A small example: let G be the two element Boolean group.  Since G satisfies x2 = 1 and, for all i and j, we have xi xj = xj xi, it's not difficult to see that there are exactly 2n distinct functions represented by the words in n variables; that is,

fnG = 2n.