EXPOSITIO

The representations of the Symmetric Groups S_n carry a very rich combinatorial structure. In particular, their representations have connections to seemingly unrelated problems, including to symmetric polynomials and the theory of Young Tableau. We demonstrate some classical res…

This paper gives an expository introduction to arithmetic progressions in Ramsey theory and additive combinatorics. We begin with the language of colorings, using small examples to illustrate how simple coloring rules can force surprisingly rigid structure. The main result discu…

Sidorenko's conjecture predicts that the homomorphism density of a bipartite graph H in a host graph G satisfies t_H(G) ≥ t_{K_2}(G)^|E(H)|. We introduce the conjecture through homomorphism densities, prove the finite-host inequality for P_3, C_4, and the complete bipartite grap…

In this paper we show that the chromatic function $P(G,t)$ is always a polynomial. We then ask a simple question: where does this polynomial equal zero? Tutte showed that two stretches of the real line, $(-\infty,0)$ and $(0,1)$, can never contain a root. Jackson showed in 1993 …

Combinatorial game theory and coding theory appear to be completely separate fields at first glance. However, they share a profound connection through structures known as lexicographic codes, or lexicodes. First introduced by Conway and Sloane in 1986, lexicodes are generated by…

In this paper, we explore matroids, which generalize properties of linear independence of vectors and cycles in graphs. We investigate properties and operations on matroids, then we look at how we can represent matroids using matrices.