The Largest Pure Partial Planes of Order 6 Have Size 25

TL;DR

This paper proves that the maximum size of a pure partial plane of order 6 is 25, classifies all such planes of this size, and employs computer-aided search combined with combinatorial and symmetry-based reductions.

Contribution

It establishes the maximum size of pure partial planes of order 6 and provides a complete classification of all planes of size 25, using novel computational and combinatorial methods.

Findings

Largest pure partial plane of order 6 has size 25

Complete classification of pure partial planes of size 25

Effective use of computer search with symmetry reduction

Abstract

In this paper, we prove that the largest pure partial plane of order 6 has size 25. At the same time, we classify all pure partial planes of order 6 and size 25 up to isomorphism. Our major approach is computer search. The search space is very large so we use combinatorial arguments to rule out some of the cases. For the remaining cases, we subdivide each search by phases and use multiple checks to reduce search space via symmetry.