The four-way intersection problem for latin squares

TL;DR

This paper investigates the possible intersection sizes of four Latin squares of a given order, completely characterizing the set of feasible intersections for large orders and partially for smaller ones.

Contribution

It completely determines the set of possible intersection sizes for four Latin squares of order at least 16, extending previous work on three Latin squares.

Findings

Complete characterization of $I^4[n]$ for $n \u2265 16$

Most elements of $I^4[n]$ identified for $n \u2264 16

Extension of intersection problem results from three to four Latin squares

Abstract

For $\mu$ given latin squares of order $n$, they have {\sf $k$ intersection} when they have $k$ identical cells and $n^2-k$ cells with mutually different entries. For each $n\geq 1$ the set of integers $k$ such that there exist $\mu$ latin squares of order $n$ with $k$ intersection is denoted by $I^{\mu}[n]$. In a paper by P. Adams et al. (2002), $I^3[n]$ is determined completely. In this paper we completely determine $I^4[n]$ for $n\geq 16$. For $n \le 16$, we find out most of the elements of $I^4[n]$.