We prove that a binary matroid with huge branch-width contains the cycle matroid of a large grid as a minor. This implies that an infinite antichain of binary matroids cannot contain the cycle matroid of a planar graph. The result also holds for any other finite field.

Branch-width, Matroids, Graph minors, Grids
Matroids, geometric lattices (msc 05B35)
Academic Press
Journal of Combinatorial Theory - Series B
Probability, Networks and Algorithms

Geelen, J, Gerards, A.M.H, & Whittle, G. (2007). Excluding a planar graph from GF(q)-representable matroids. Journal of Combinatorial Theory - Series B, 97, 971–998.