Lam's Problem

In finite geometry, Lam's problem is the problem of determining if a finite projective plane of order ten exists.
The order ten case is the first theoretically uncertain case, as all smaller orders can be resolved by purely theoretical means.
Lam's problem is named after Clement W. H. Lam who experimentally determined that projective planes of order ten do not exist via exhaustive computational searches.

Introduction

A finite projective plane of order $n$ is a collection of points and lines such that
* any two points define a unique line,
* any two lines meet at a unique point,
* there are exactly $n+1$ points on every line, and
* there are exactly $n+1$ lines through every point.
A consequence of this definition is that a projective plane of order $n$ will contain $n^2+n+1$ points and $n^2+n+1$ lines.
The incidence relation between points and lines may equivalently be described using an incidence matrix. In this context a projective plane of order $n$
is equivalent to a $(n^2+n+1)\times(n^2+n+1)$ matrix with $\$ entries such that every row and column has $n+1$ ones
and the inner product between any two rows or columns is exactly $1$.

Using the incidence matrix representation, Lam's problem is equivalent to determining if there is a way of placing 0s and 1s in a $111\times111$ matrix such that there are exactly eleven 1s in each row and column and any pair of
rows share a single 1 in the same column.

Clement W. H. Lam considered studying the existence of a projective plane of order ten in his PhD thesis but was dissuaded by his thesis advisor H. J. Ryser

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