Ramsey numbers, denoted as R(s,t), are fundamental in graph theory, representing the smallest number of vertices n such that every graph on n vertices either contains a clique of size s or an independent set of size t. Recent developments in Ramsey theory have focused on finding asymptotic bounds for Ramsey numbers. In this talk, we survey asymptotic bounds of Ramsey Numbers R(3,t) and R(4,t),...
We present extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes, providing a common generalization to the topological colorful Helly theorem by Kalai and Meshulam, the very colorful Helly theorem by Arocha et al., and the semi-intersecting colorful Helly theorem by Karasev and Montejano. As an application, we obtain a strengthened version of Tverberg's theorem. This...
The 1-skeleton of a convex polytope $P$ is called the graph of $P$.
A graph of a simplicial $d$-polytope is said to be toric colorable if there is a vertex coloring $\lambda \colon V(G) \to \mathbb{Z}^d$ such that $\{v_1, \ldots, v_d\}$ forms a face of $P$ implies that $\{\lambda(v_1), \ldots, \lambda(v_d)\}$ is unimodular.
In this talk, we discuss the toric colorability of graphs of...
Lusztig $q$ weight multiplicity is a polynomial in $q$ whose positivity has been verified by linking it to a specific affine Kazhdan-Lusztig polynomial. However, a combinatorial formula beyond type A has not been known until recently.
In 2019, Lee proposed a combinatorial formula for type C using a novel combinatorial concept known as semistandard oscillating tableaux. We will outline the...
In 2019, Butler, Choi, Kim, and Seo introduced a new type of juggling card that represents multiplex juggling patterns in a natural bijective way. They conjectured a formula for the generating function for the number of multiplex juggling cards with capacity.
In this paper we prove their conjecture. More generally, we find an explicit formula for the generating function with any capacity....
