KPPY 86

A group $G$ is called residually finite if for each $1 \neq g \in G$, there is a finite group $H$ and a homomorphism $\varphi : G \rightarrow H$ such that $\varphi(g) \neq_H 1$. It is a famous open problem whether every hyperbolic group is residually finite, and is commonly believed that a non-residually finite hyperbolic group exists.

Relatively hyperbolic groups which satisfy small cancellation conditions are closely related to hyperbolic groups. We construct a non-residually finite direct limit which satisfies small cancellation conditions $C(4)\&T(4)$, and consists of finitely presented relatively hyperbolic groups by using the upper presentations of 2-bridge link groups.

The Smith normal form in design theory is a useful invariant to distinguish designs of the same order.

Recently Armario partially determined the Smith normal form of a skew-symmetric D-optimal design of order $n\equiv 2\pmod{4}$, and conjectured the remaining invariant factors.

In this talk, we prove his conjecture. This talk is based on joint work with Gary Greaves.

We say that a graph class C satisfies the Erdos-Posa property if there exists a function g only depending on C such that for every graph G and an integer k, either G contains k disjoint copies of graphs in C, or it contains a vertex set of size at most g(k) that hits all copies of graphs in C. We recently determined for many classes C whether C satisfies Erdos-Posa property or not for induced version. There are still many open problems, and we will see these in the talk.

We show non-existence of $Q$-polynomial association schemes in some small open cases. To do so, we analyse certain linear Diophantine equations involving their triple intersection numbers to conclude that they are inconsistent. This approach was previously applied to $P$-polynomial association schemes (i.e., distance-regular graphs) with vanishing some of their Krein parameters.

The talk is based on joint work (in progress) with Jano\v{s} Vidali.

A celebrated conjecture of Gyárfás and Sumner asserts that, for a forest $F$ and a complete graph $K$, every graph not containing $F$ or $K$ as an induced subgraph has bounded chromatic number. A similar question for tournaments (a digraph whose underlying graph is a complete graph) can be asked. For which set $\mathcal{H}$ of tournaments, does every tournament not containing any member of $\mathcal{H}$ have bounded chromatic number? Such a set containing one tournament has been explicitly characterized by Berger et al. In this talk, we discuss such sets containing two tournaments.