# KPPY 84

Sang June Lee, Mitsugu Hirasaka, Norihide Tokushige, Kang-Ju Lee, Jongyook Park

Sept 16, 2017. 11am - 5:30pm at KNU

Schedule | ||

11:00am - 11:50 |
Sang June Lee Duksung Women's University | Infinite Sidon sets contained in sparse random sets of integers |

12pm - 1:25 | Lunch | |

1:30 - 2:20 |
Mitsugu Hirasaka Pusan University | On isometric sequences of colored spaces |

2:30 - 3:20 |
Norihide Tokushige University of the Ryukyus, Japan | The maximum product of measures of cross t-intersecting families |

3:30 - 4:20 |
Kang-Ju Lee Seoul Nataional University | Simplicial networks and effective resistance |

4:30 - 5:20 |
Jongyook Park Won-kwang University | On the number of vertices for non-antipodal distance-regular graphs |

## Abstracts

Sang June Lee

Infinite Sidon sets contained in sparse random sets of integers

Infinite Sidon sets contained in sparse random sets of integers

A set $S$ of natural numbers is a

This is joint work with Y. Kohayakawa, C. G. Moreira and V. Rödl.

*Sidon*set if all the sums $s_1+s_2$ with $s_1$, $s_2\in S$ and $s_1\leq s_2$ are distinct. Let constants $\alpha>0$ and $0<\delta<1$ be fixed, and let $p_m=\min\{1,\alpha m^{-1+\delta}\}$ for all positive integers $m$. Generate a random set $R\subset\mathbb{N}$ by adding $m$ to $R$ with probability $p_m$, independently for each $m$. We investigate how dense a Sidon set $S$ contained in $R$ can be. Our results show that the answer is qualitatively very different in at least three ranges of $\delta$. We prove quite accurate results for the range $0<\delta\leq2/3$, but only obtain partial results for the range $2/3<\delta\leq1$.This is joint work with Y. Kohayakawa, C. G. Moreira and V. Rödl.

Mitsugu Hirasaka

On isometric sequences of colored spaces

On isometric sequences of colored spaces

A

In this talk we aim to classify colored spaces with $a_2(r)=a_3(r)$.

This is a joint work with Masashi Shinohara.

*colored space*is the pair of a set $X$ and a function $r$ whose domain is $\binom{X}{2}$. Let $(X,r)$ be a finite colored space and $Y,Z\subseteq X$. We shall write $Y\simeq_r Z$ if there exists a bijection $f:Y\to Z$ such that $r(U)=r(f(U))$ for each $U\in\binom{Y}{2}$. Notice that, for $U,V\in \binom{X}{2}$, $U\simeq_r V$ if and only if $r(U)=r(V)$, and for $Y,Z\in \binom{X}{3}$, $Y\simeq_r Z$ if and only if $(r(U)\mid U\in \binom{Y}{2})$ is a replacement of $(r(V)\mid V\in \binom{Z}{2})$. We denote the numbers of equivalence classes contained in $\binom{X}{2}$ and $\binom{X}{3}$ by $a_2(r)$ and $a_3(r)$, respectively.In this talk we aim to classify colored spaces with $a_2(r)=a_3(r)$.

This is a joint work with Masashi Shinohara.

Norihide Tokushige

The maximum product of measures of cross t-intersecting families

The maximum product of measures of cross t-intersecting families

$
\def\cA{\mathcal A}
\def\cB{{\mathcal B}}
\def\cF{{\mathcal F}}
$
For a positive integer $n$ let $[n]:=\{1,2,\ldots,n\}$ and let
$\Omega:=2^{[n]}$ denote the power set of $[n]$.
A family of subsets ${\mathcal A}\subset \Omega$ is called $t$-intersecting
if $|A\cap A'|\geq t$ for all $A,A'\in{\mathcal A}$.
Let $p\in(0,1)$ be a fixed real number. We define the product measure
$\mu:2^{\Omega}\to[0,1]$ by $\mu(\cA):=\sum_{A\in\cA}p^{|A|}(1-p)^{n-|A|}$
for $\cA\in 2^{\Omega}$.
Ahlswede and Khachatrian proved that if
\begin{equation*}
\frac r{t+2r-1}\leq p\leq \frac{r+1}{t+2r+1}
\end{equation*}
and $\cA\subset\Omega$ is $t$-intersecting, then $\mu(\cA)\leq\mu(\cF^t_r)$,
where $\cF_r^t$ is a $t$-intersecting family defined by
$\cF_r^t:=\{F\subset[n]:|F\cap[t+2r]|\geq t+r\}$.

We extend this result to two families. We say that two families $\cA,\cB\subset\Omega$ are cross $t$-intersecting if $|A\cap B|\geq t$ for all $A\in\cA,B\in\cB$. In this case it is conjectured that $\mu(\cA)\mu(\cB)\leq\mu(\cF^t_r)^2$ for $p$ in the range given above. In my talk I will report that this conjecture is true if $t\gg r$. I will also discuss a related stability result.

This is joint work with Sang June Lee and Mark Siggers.

We extend this result to two families. We say that two families $\cA,\cB\subset\Omega$ are cross $t$-intersecting if $|A\cap B|\geq t$ for all $A\in\cA,B\in\cB$. In this case it is conjectured that $\mu(\cA)\mu(\cB)\leq\mu(\cF^t_r)^2$ for $p$ in the range given above. In my talk I will report that this conjecture is true if $t\gg r$. I will also discuss a related stability result.

This is joint work with Sang June Lee and Mark Siggers.

Kang-Ju Lee

Simplicial networks and effective resistance

Simplicial networks and effective resistance

We introduce the notion of effective resistance for a

The effective resistance $R_{\sigma}$ of a current generator $\sigma$ shall be defined as a ratio of the $\sigma$-components of $V_{\sigma}$ and $I_{\sigma}$. By introducing

This is a joint work with Woong Kook.

*simplicial network*$(X,R)$ where $X$ is a simplicial complex and $R$ is a set of resistances for the top simplices, and prove two formulas generalizing previous results concerning effective resistance for resistor networks. Our approach, based on combinatorial Hodge theory, is to assign a unique harmonic class to a*current generator*$\sigma$, an extra top-dimensional simplex to be attached to $X$. We will show that the harmonic class gives rise to the*current*$I_{\sigma}$ and the*voltage*$V_{\sigma}$ for $X\cup\sigma$, satisfying Thompson's energy-minimizing principle and Ohm's law for simplicial networks.The effective resistance $R_{\sigma}$ of a current generator $\sigma$ shall be defined as a ratio of the $\sigma$-components of $V_{\sigma}$ and $I_{\sigma}$. By introducing

*potential*for voltage vectors, we present a formula for $R_\sigma$ via the inverse of the weighted combinatorial Laplacian of $X$ in codimension one. We also derive a formula for $R_{\sigma}$ via weighted high-dimensional tree-numbers for $X$, providing a combinatorial interpretation for $R_{\sigma}$. As an application, we generalize Foster's Theorem, and discuss various high-dimensional examples.This is a joint work with Woong Kook.

Jongyook Park

On the number of vertices for non-antipodal distance-regular graphs

On the number of vertices for non-antipodal distance-regular graphs

Let $\Gamma$ be a distance-regular graph with valency $k$ and diameter $D$, and let $x$ be a vertex of $\Gamma.$
We denote by $k_i$ $(0\leq i \leq D)$ the number of vertices at distance $i$ from $x$. In this talk, we try to quantify the difference between antipodal and non-antipodal distance-regular graphs. We will look at the sum $k_{D-1} + k_D$, and consider the situation where $k_{D-1}+k_D\leq 2k$. If $\Gamma$ is an antipodal distance-regular graph, then $k_{D-1} + k_{D} = k_D (k+1)$. It follows that either $k_D =1$ or the graph is non-antipodal. And for a non-antipodal distance-regular graph, it was known that $k_D(k_D-1) \geq k$ and $k_{D-1} \geq k$ both hold. So, this talk concerns on obtaining more detailed information on the number of vertices for a non-antipodal distance-regular graph. We first concentrate on the case where the diameter equals three. In this case, the condition $k_D + k_{D-1} \leq 2k$ is equivalent to the condition that the number of vertices is at most $3k+1$. And we extend this result to all diameters. We note that although the result of the diameter $3$ case is a corollary of the result of all diameters, the main difficulty is the diameter $3$ case, and that the diameter $3$ case confirms the following conjecture: there is no primitive distance-regular graph with diameter $3$ having the $M$-property.