The critical number of dense triangle-free binary matroids

We show that, for each real number ε>0ε>0 there is an integer c such that, if M   is a simple triangle-free binary matroid with |M|≥(14+ε)2r(M), then M has critical number at most c  . We also give a construction showing that no such result holds when replacing 14+ε with 14−ε in this statement. This shows that the “critical threshold” for the triangle is 14. We extend the notion of critical threshold to every simple binary matroid N and conjecture that, if N   has critical number c≥3c≥3, then N   has critical threshold 1−i⋅2−c1−i⋅2−c for some i∈{2,3,4}i∈{2,3,4}. We give some support for the conjecture by establishing lower bounds.