The 84th KPPY Combinatorics Seminar
Organized by S.Bang, M.Hirasaka, T.Jensen, and M.Siggers Sept 16, 2017
Department of Mathematics, Kyungpook National University Natural Sciences Building (Building 209) , Room 313
Program
11:00 - 11:50 Sang June Lee Duksung Women’s University Infinite Sidon sets contained in sparse random sets of integers 12:00 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 3:40 - 4:30 Kang-Ju LeeSeoul Nataional University
Simplicial networks and effective resistance
4:40 - 5:30 Jongyook Park Won-kwang University
On the number of vertices for non-antipodal distance-regular graphs 5:30 - 8:30 Banquet
Abstracts
Sang June Lee
Infinite Sidon sets contained in sparse random sets of integers
A set S of natural numbers is a Sidon set if all the sums s1 +s2 with s1, s2 ∈S and s1 ≤s2 are distinct. Let constantsα >0 and 0< δ <1 be fixed, and letpm = min{1, αm−1+δ}for all positive integersm. Generate a random setR ⊂Nby addingm toR with probabilitypm, 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 δ. We prove quite accurate results for the range 0 < δ ≤ 2/3, but only obtain partial results for the range 2/3< δ ≤1.
This is joint work with Y. Kohayakawa, C. G. Moreira and V. R¨odl.
Mitsugu Hirasaka
On isometric sequences of colored spaces
A colored space is the pair of a setX and a function r whose domain is X2 . Let (X, r) be a finite colored space and Y, Z ⊆ X. We shall write Y 'r Z if there exists a bijection f : Y → Z such that r(U) = r(f(U)) for each U ∈ Y2
. Notice that, for U, V ∈ X2
, U 'r V if and only if r(U) = r(V), and for Y, Z ∈ X3
, Y 'r Z if and only if (r(U)|U ∈ Y2
) is a replacement of (r(V)|V ∈ Z2
). We denote the numbers of equivalence classes contained in X2
and X3
by a2(r) and a3(r), respectively.
In this talk we aim to classify colored spaces with a2(r) = a3(r).
This is a joint work with Masashi Shinohara.
Norihide Tokushige
The maximum product of measures of cross t-intersecting families For a positive integer n let [n] := {1,2, . . . , n} and let Ω := 2[n] denote the power set of [n]. A family of subsets A ⊂ Ω is called t-intersecting if
|A∩A0| ≥ t for all A, A0 ∈ A. Let p ∈ (0,1) be a fixed real number. We
define the product measure µ: 2Ω →[0,1] by µ(A) :=P
A∈Ap|A|(1−p)n−|A|
for A ∈2Ω. Ahlswede and Khachatrian proved that if r
t+ 2r−1 ≤p≤ r+ 1 t+ 2r+ 1
and A ⊂Ω is t-intersecting, thenµ(A)≤µ(Frt), where Frt is a t-intersecting family defined by Frt:={F ⊂[n] :|F ∩[t+ 2r]| ≥t+r}.
We extend this result to two families. We say that two families A,B ⊂ Ω are cross t-intersecting if |A∩B| ≥t for all A∈ A, B ∈ B. In this case it is conjectured that µ(A)µ(B)≤ µ(Frt)2 for p in the range given above. In my talk I will report that this conjecture is true if t 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
We introduce the notion of effective resistance for asimplicial network (X, R) whereX is a simplicial complex andR is a set of resistances for the top sim- plices, and prove two formulas generalizing previous results concerning effec- tive resistance for resistor networks. Our approach, based on combinatorial Hodge theory, is to assign a unique harmonic class to a current generator σ, an extra top-dimensional simplex to be attached to X. We will show that the harmonic class gives rise to the current Iσ and the voltage Vσ for X∪σ, satisfying Thompson’s energy-minimizing principle and Ohm’s law for sim- plicial networks.
The effective resistance Rσ of a current generator σ shall be defined as a ratio of the σ-components ofVσ and Iσ. By introducing potential for voltage vectors, we present a formula for Rσ via the inverse of the weighted combi- natorial Laplacian of X in codimension one. We also derive a formula forRσ via weighted high-dimensional tree-numbers for X, providing a combinato- rial interpretation forRσ. 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 Let Γ be a distance-regular graph with valency k and diameter D, and let x be a vertex of Γ. We denote by ki (0 ≤ i ≤ 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 kD−1 +kD, and consider the situation where kD−1+kD ≤ 2k. If Γ is an antipodal distance-regular graph, then kD−1+kD =kD(k+ 1). It follows that either kD = 1 or the graph is non-antipodal. And for a non-antipodal distance-regular graph, it was known that kD(kD −1) ≥ k and kD−1 ≥ 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 kD +kD−1 ≤ 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.
