# The minimum possible volume size of u-way (v, k, t) trades

## TL;DR

This paper investigates the smallest possible size and foundation of u-way (v, k, t) trades, specifically determining these parameters for the case when t=2 and k=t+1, for any u > 2.

## Contribution

It provides the first comprehensive determination of the minimum volume and foundation size for u-way (v, t+1, t) trades when t=2, for all u > 2.

## Key findings

- Minimum volume of u-way (v, t+1, t) trades established for t=2.
- Minimum foundation size determined for these trades.
- Results applicable for all u > 2.

## Abstract

A u-way (v; k; t) trade is a pair T = (X; T_1; T2,...,T_u) such that for each t-subset of v-set X the number of blocks containing this t-subset is the same in each Ti (1 <= i <=u). In the other words for each 1 <= i < j <= u, (X; T_i; T_j) is a (v; k; t) trade. There are many questions concerning u-way trades. The main question is about the minimum volume and minimum foundation size of u-way (v; k; t) trades. In this paper, we determine the minimum volume and minimum foundation size of u-way (v; t + 1; t) trades for each integer number u >2 and t = 2.