Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The critical threshold of a (simple binary) matroid N is the infimum over all ρ such that any N-free matroid M   with |M|>ρ2r(M)|M|>ρ2r(M) has bounded critical number. ,  resolve two conjectures of Geelen and Nelson, showing that the critical threshold of the projective geometry PG(t−1,2)PG(t−1,2) is 1−3⋅2−t1−3⋅2−t.  do so by proving the following stronger statement: if M   is PG(t−1,2)PG(t−1,2)-free with |M|>(1−3⋅2−t)2r(M)|M|>(1−3⋅2−t)2r(M), then the critical number of M   is t−1t−1 or t. Together with earlier results of Geelen and Nelson [9] and Govaerts and Storme [11], this completes the classification of dense PG(t−1,2)PG(t−1,2)-free matroids.
['Jonathan Tidor']
I am writing a research paper on a idea Dense binary $PG(t-1,2)$-free matroids have critical number $t-1$ or $t$
[{"from": "human", "value": "I am writing a research paper on a idea Dense binary $PG(t-1,2)$-free matroids have critical number $t-1$ or $t$"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The critical threshold of a (simple binary) matroid N is the infimum over all \u03c1 such that any N-free matroid M \u00a0 with |M|>\u03c12r(M)|M|>\u03c12r(M) has bounded critical number. ,  resolve two conjectures of Geelen and Nelson, showing that the critical threshold of the projective geometry PG(t\u22121,2)PG(t\u22121,2) is 1\u22123\u22c52\u2212t1\u22123\u22c52\u2212t.  do so by proving the following stronger statement: if M \u00a0 is PG(t\u22121,2)PG(t\u22121,2)-free with |M|>(1\u22123\u22c52\u2212t)2r(M)|M|>(1\u22123\u22c52\u2212t)2r(M), then the critical number of M \u00a0 is t\u22121t\u22121 or t. Together with earlier results of Geelen and Nelson [9] and Govaerts and Storme [11], this completes the classification of dense PG(t\u22121,2)PG(t\u22121,2)-free matroids."}]