A Constructive Approach to the Critical Problem for Matroids

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Let M be a loopless matroid on a finite non-empty set I which is representable over a finite field F . Let A be a representation of M over F , i.e. A is a family ( A i , i ∈ I ) indexed by I of vectors of F n (for some n ⩾ 1) such that J ⊆ I is an independent set of M iff ( A i , i ∈ J ) is a linearly independent family of vectors of F n . The critical number of A, c (A) , is the smallest integer k ⩾ 1 such that there exist k linear functionals f j : F n → F ( j ∈ {1, ... , k }) with the following property: ∀ i ∈ I , ∃ j ∈ {1, ... , k } such that f j ( A i ) ≠ 0. It is known that all representations of M over F have the same critical number: by definition this number is the critical number of M over F and is denoted by c F ( M ). The critical problem (determine c F ( M ) for a given finite field F and for a given class of matroids M representable over F ) contains as special cases some of the most fascinating problems in graph theory. We present here, for any k ⩾ 2 and finite field F , a simple constructive characterization of matroids M representable over F with c F ( M ) ⩾ k . Our result is quite similar to the Theorem of Hajós which gives a constructive characterization of simple graphs with chromatic number at least q (∀ q ⩾3).