Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5777625 | Journal of Combinatorial Theory, Series B | 2017 | 15 Pages |
Abstract
The critical threshold of a (simple binary) matroid N is the infimum over all Ï such that any N-free matroid M with |M|>Ï2r(M) has bounded critical number. In this paper, we resolve two conjectures of Geelen and Nelson, showing that the critical threshold of the projective geometry PG(tâ1,2) is 1â3â
2ât. We do so by proving the following stronger statement: if M is PG(tâ1,2)-free with |M|>(1â3â
2ât)2r(M), then the critical number of M is tâ1 or t. Together with earlier results of Geelen and Nelson [9] and Govaerts and Storme [11], this completes the classification of dense PG(tâ1,2)-free matroids.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Jonathan Tidor,