On the impossibility of decomposing binary matroids

We show that there exist k-colorable matroids are not (b,c)-decomposable when b and c constants. A matroid is (b,c)-decomposable, if its ground set of elements can be partitioned into sets X1,X2,…,Xℓ with the following two properties. Each Xi has size at most ck. Moreover, for all Y such |Y∩Xi|≤1 it case b-colorable. (b,c)-decomposition a strict generalization partition decomposition and, thus, our result refutes conjecture from [4].