Rota’s basis conjecture holds for random bases of vector spaces

Abstract

In 1989, Rota conjectured that, given n bases B1,…,Bn of the vector space Fn over some field F, one can always decompose the multi-set B1∪⋯∪Bn into transversal bases. This conjecture remains wide open despite of a lot of attention. In this paper, we consider the setting of random bases B1,…,Bn. More specifically, our first result shows that Rota’s basis conjecture holds with probability 1−o(1) as n→∞ if the bases B1,…,Bn are chosen independently uniformly at random among all bases of Fqn for some finite field Fq (the analogous result is trivially true for an infinite field F). In other words, the conjecture is true for almost all choices of bases B1,…,Bn⊆Fqn. Our second, more general, result concerns random bases B1,…,Bn⊆Sn for some given finite subset S⊆F (in other words, bases B1,…,Bn where all vectors have entries in S). We show that when choosing bases B1,…,Bn⊆Sn independently uniformly at random among all bases that are subsets of Sn, then again Rota’s basis conjecture holds with probability 1−o(1) as n→∞.