We show that, for each real number epsilon > 0 there is an integer c such that, if M is a simple triangle-free binary matroid with vertical bar M vertical bar >= (1/4+epsilon)2(r(M)), then M has critical number at most c. We also give a construction showing that no such result holds when replacing 1/4 + epsilon with 1/4 - epsilon in this statement. This shows that the "critical threshold" for the triangle is 1/4. We extend the notion of critical threshold to every simple binary matroid N and conjecture that, if N has critical number c >= 3, then N has critical threshold 1 - i . 2(-c) for some i is an element of {2, 3, 4}. We give some support for the conjecture by establishing lower bounds. (C) 2015 Elsevier Inc. All rights reserved.
展开▼
