# Two short combinatorics talks

## ABSTRACT

There will be two 30 min. talks with the following titles and abstracts:

## Identities of Ehrhart polynomials arising from algebraic geometry, by Fu Liu

Mochizuki's work on torally indigenous bundles yields combinatorial identities by generating to different curves of the same genus. We rephrase these identities in combinatorial languages and strengthen them. The main result of our work is that given a certain way to construct a polytope from a connected trivalent graph, the odd values of the Ehrhart quasi-polynomials of this polytope is a single polynomial which depends only on the number of vertices and edges of the original trivalent graph. Then we are able to conclude that the number of dormant torally indigenous bundles on a general curve of a given type is expressed as a polynomial in the characteristic of the base field.

This is a joint work with Brian Osserman.

## Ramsey Theory on the Integers, by Jacob Fox

In this talk, I will present classical results and new developments in Ramsey theory on the integers.

A system of linear equations is called partition k-regular if for every k-coloring of the positive integers, there exists a monochromatic solution to the given system of linear equations. A system of linear equations is called partition regular if it is partition k-regular for all k. Generalizing classical theorems of Schur and van der Waerden, Richard Rado classified the finite partition regular system of linear equations in his famous 1933 dissertation Studien zur Kombinatorik. Rado further conjectured that for all positive integers m, n, there exists k=k(m,n) such that every system of m linear equations in n variables that is partition k-regular must be partition regular.

Professor Daniel Kleitman and I recently proved the first nontrivial case, when n=3, of this conjecture, which is known as Rado's Boundedness Conjecture. In particular, if a, b, and c are fixed rational numbers, and if every 6-coloring of the nonzero rational numbers must have a monochromatic solution to ax + by + cz = 0, then every finite coloring of the positive integers must have a monochromatic solution to ax + by + cz = 0. I will discuss this result, as well as several recent conjectures whose proofs follow from our research. No prior knowledge of Ramsey theory is assumed.

Speaker's Contact Info: fuliu(at-sign)math.mit.edu, licht(at-sign)mit.edu