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),...

A sequence $(e_1, e_2, \cdots, e_n)$ is an inversion sequences if $0\leq e_i < i$ for all $i=1, \ldots, n$. We say that an inversion sequences $e=(e_1, e_2, \cdots, e_n)$ \emph{contains} the pattern $102$ if there exist some indices $i < j < k$ such that $e_j < e_i < e_k$. Otherwise, $e$ is said to \emph{avoid} the pattern $102$.

In this talk, we will construct a correspondence between the...

For a given hypergraph $H$ and a vertex $v\in V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H$. In 1995, Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $v\in V(H)$. This conjecture was proved for $k = 2$ by Kahn and Kim in 1998.

We...

Let $\Gamma$ be a graph with diameter at least two. Then $\Gamma$ is said to be $1$-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $\Gamma$, the distance partition of the vertex set of $\Gamma$ with respect to both $x$ and $y$ is equitable, and the parameters corresponding to equitable partitions are independent of the choice of $x$ and $y$....

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...

It is well-known that a tournament always contains a directed Hamilton path. Rosenfeld conjectured that if a tournament is sufficiently large, it contains a Hamilton path of any given orientation. This conjecture was approved by Thomason, and Havet and Thomassé completely resolved it by showing there are exactly three exceptions.

We generalized this result into a transversal setting. Let...

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...

In this talk, we focus on the rational Betti numbers of real toric manifolds associated with chordal nestohedra. We introduce an explicit description for the Betti numbers using alternating $\mathcal{B}$-permutations for a chordal building set $\mathcal{B}$. We provide detailed computations for interesting cases of chordal nestohedra, including permutohedra, associahedra, stellohedra,...

Bott manifolds are smooth projective toric varieties providing interesting avenues among topology, geometry, representation theory, and combinatorics. They are used to understand the geometric structure of Bott-Samelson-Demazure-Hansen (BSDH) varieties, which provide desingularizations of Schubert varieties. However, not all Bott manifolds originate from BSDH varieties. Those that do are...

The Vietoris-Rips complex, originally introduced by Leopold Vietoris in the early 1900s to develop a homology theory for metric spaces, has since found applications in various areas of mathematics. Eliyahu Rips and Mikhail Gromov further utilized it in their studies of hyperbolic groups. More recently, classifying the homotopy types of Vietoris-Rips complexes has become a significant problem...

The $(k,r)$-incidence graph of a regular polytope $\mathcal{P}$ is the bipartite incidence graph between $k$-faces and $r$-faces of $\mathcal{P}$. We obtain a general upper bound and a corresponding supersaturation result for the extremal number of the $(k,r)$-incidence graph of any regular polytope.

This generalises recent results of Janzer and Sudakov, who obtained the same bound for...

Orthogonal polynomials on the unit circle (OPUC) are a family of polynomials orthogonal with respect to integration on the unit circle in the complex plane. The values of these integrals can be obtained by calculating moments. Numerous combinatorial studies have explored the moments of various types of orthogonal polynomials, including classical orthogonal polynomials, Laurent biorthogonal...

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....

Dress (1986) introduced matroids with coefficients offering a unified approach to ordinary matroids, representations of matroids over fields, and oriented matroids. Baker and Bowler (2019) extended this theory, whose result includes a partial field representation by Semple and Whittle (1996).

I will present two generalizations of matroids with coefficients. One is about skew-symmetric...