Stability and exact Turán numbers for matroids

Abstract

We consider the Turán-type problem of bounding the size of a set M ⊆ F 2 n that does not contain a linear copy of a given fixed set N ⊆ F 2 k , where n is large compared to k . An Erdős-Stone type theorem [5] in this setting gives a bound that is tight up to a o ( 2 n ) error term; our first main result gives a stability version of this theorem, showing that such an M that is close in size to the upper bound in [5] is close to the obvious extremal example in the sense of symmetric difference . Our second result shows that the error term in [5] is exactly controlled by the solution to one of a class of ‘sparse’ extremal problems, and gives some examples where the error term can be eliminated completely to give a sharp upper bound on | M | .