Scrambled Rota conjecture

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Let [math]M=(S,r)[/math] be a loopless matroid of rank k whose ground set can be partitioned into k bases. Is it true that no matter how we partition S into sets of size k, the partition will have k-1 disjoint transversals that are bases?


This was proposed by Aharoni and Kotlar [1]. They give a simple example showing that k-1 cannot be replaced by k in the conjecture. A somewhat similar problem is the special case of Upper bound on common independent set cover when one of the matroids is a partition matroid. See also the survey of Aharoni, Charbit and Howard [2] that contains several related questions.


  1. R. Aharoni, D. Kotlar, A weak version of Rota's basis conjecture for odd dimensions, DOI link, arXiv link
  2. R. Aharoni, P. Charbit and D. Howard, On a Generalization of the Ryser-Brualdi-Stein Conjecture, DOI link, arXiv link