The 3-way intersection problem for S(2, 4, v) designs

TL;DR

This paper investigates the 3-way intersection problem for S(2,4,v) designs, establishing bounds and exact values for the number of common blocks in three such designs for various v, including complete solutions for specific cases.

Contribution

The paper determines the set of possible common block counts for three S(2,4,v) designs, proving bounds and exact values for many v, extending previous knowledge in combinatorial design theory.

Findings

J_3[v] is a subset of I_3[v] for v ≡ 1,4 (mod 12)

J_3[v] equals I_3[v] for v ≥ 49 and v=13

Complete determination of J_3[16] and partial results for v=25,28,37,40

Abstract

In this paper the 3-way intersection problem for $S(2,4,v)$ designs is investigated. Let $b_{v}=\frac {v(v-1)}{12}$ and $I_{3}[v]=\{0,1,...,b_{v}\}\setminus\{b_{v}-7,b_{v}-6,b_{v}-5,b_{v}-4,b_{v}-3,b_{v}-2,b_{v}-1\}$. Let $J_{3}[v]=\{k|$ there exist three $S(2,4,v)$ designs with $k$ same common blocks$\}$. We show that $J_{3}[v]\subseteq I_{3}[v]$ for any positive integer $v\equiv1, 4\ (\rm mod \ 12)$ and $J_{3}[v]=I_{3}[v]$, for $ v\geq49$ and $v=13 $. We find $J_{3}[16]$ completely. Also we determine some values of $J_{3}[v]$ for $\ v=25,28,37$ and 40.