On the existence of $d$-homogeneous $3$-way Steiner trades

TL;DR

This paper characterizes 3-way 3-homogeneous Steiner trades of volume v and provides constructions for certain values of d, advancing the understanding of combinatorial trade structures.

Contribution

It offers a complete characterization of 3-way 3-homogeneous (v,3,2) Steiner trades and constructs for specific d values, filling gaps in existing combinatorial trade theory.

Findings

Characterization of 3-way 3-homogeneous (v,3,2) Steiner trades

Construction methods for 3-way d-homogeneous trades with d in {4,5,6}

Identification of seven small v values where constructions are not provided

Abstract

A $\mu$-way $(v, k, t)$ trade $T = \{T_{1} , T_{2}, . . ., T_{\mu} \}$ of volume $m$ consists of $\mu$ disjoint collections $T_{1}, T_{2}, \ldots, T_{\mu}$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$ the number of blocks containing this $t$-subset is the same in each $T_{i}$ (for $1 \leq i \leq \mu$). A $\mu$-way $(v, k, t)$ trade is called $\mu$-way $(v, k, t)$ Steiner trade if any $t$-subset of found$(T)$ occurs at most once in $T_{1}$ $(T_{j},\ j \geq 2)$. A $\mu$-way $(v,k,t)$ trade is called $d$-homogeneous if each element of $V$ occurs in precisely $d$ blocks of $T_{1}$ $(T_{j},~ j \geq 2)$. In this paper we characterize the $3$-way $3$-homogeneous $(v,3,2)$ Steiner trades of volume $v$. Also we show how to construct a $3$-way $d$-homogeneous $(v,3,2)$ Steiner trade for $d\in \{4,5,6\}$, except for seven small values of $v$.