# KPPY 86

Jan Kim, Sho Suda, O-joung Kwon, Alexander Gavrilyuk, Ringi Kim

May 04, 2018 (Friday). 11:00am-6:00pm at PNU

Schedule | ||

11:00am - 11:50 |
Jan Kim Pusan National University | On the Construction of Non-residually Finite Direct Limit of Finitely Presented Groups |

12pm - 1:25 | Lunch | |

1:30 - 2:20 |
Sho Suda Aichi University of Education | On the Smith normal form of skew-symmetric D-optimal designs of order $n\equiv2\pmod{4}$ |

2:30 - 3:20 |
O-joung Kwon Incheon National University | Erdos-Posa property of induced objects |

3:30 - 4:20 |
Alexander Gavrilyuk Pusan National University | Non-existence of some small $Q$-polynomial association schemes |

4:30 - 5:20 |
Ringi Kim KAIST | Subtournaments in Tournaments with large chromatic number |

## Abstracts

Jan Kim

On the Construction of Non-residually Finite Direct Limit of Finitely Presented Groups

On the Construction of Non-residually Finite Direct Limit of Finitely Presented Groups

A group $G$ is called

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.

*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.

Sho Suda

On the Smith normal form of skew-symmetric D-optimal designs of order $n\equiv2\pmod{4}$

On the Smith normal form of skew-symmetric D-optimal designs of order $n\equiv2\pmod{4}$

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.

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.

O-joung Kwon

Erdos-Posa property of induced objects

Erdos-Posa property of induced objects

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.

Alexander Gavrilyuk

Non-existence of some small $Q$-polynomial association schemes

Non-existence of some small $Q$-polynomial association schemes

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.

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

Ringi Kim

Subtournaments in Tournaments with large chromatic number

Subtournaments in Tournaments with large chromatic number

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.