Rota's basis conjecture
This very pretty conjecture of Rota has a number of interesting extensions. One (obvious) extension is to matroids, and indeed, Rota conjectured this generalization (just replace the vector space of dimension by a matroid of rank ). This generalization has received some recent attention: Geelen and Humphries [GH] proved the special case when is a paving matroid, Geelen and Webb [GW] proved that in general there exist disjoint transversals which are bases, and Aharoni and Berger [AB] used their beautiful "intersection of a matroid and a simplicial complex" theorem to prove that the union of the bases can be partitioned into partial independent transversals. Another generalization of Rota's basis conjecture is the following.
To the best of my (M. DeVos') knowledge, this conjecture may generalize to matroids as well. There is another conjecture which implies Rota's basis conjecture which is far less obvious. In fact, Rota's conjecture (for even and characteristic zero) is implied by Alon and Tarsi's Even vs. odd latin squares conjecture. This implication was first discovered by Rota and Huang [RH] (Rota had himself made another conjecture equivalent to that of Alon and Tarsi), and a very transparent proof of this fact based on a polynomial identity is given by Shmuel Onn [O].
Thanks to this last implication, partial results on the Alon-Tarsi conjecture show that Rota's basis conjecture holds whenever has the form or where is an odd prime [D1], [D2], [Z].
[AB] R. Aharoni, E. Berger, The intersection of a matroid and a simplicial complex. Trans. Amer. Math. Soc. 358 (2006), no. 11, 4895--4917 MathSciNet.
[D1] A. Drisko, On the number of even and odd Latin squares of order , Adv. Math. 128 (1997), no. 1, 20--35. MathSciNet
[GW] J. Geelen, and K. Webb, On Rota's basis conjecture
*[HR] R. Huang and G-C Rota, On the relations of various conjectures on Latin squares and straightening coefficients. Discrete Math. 128 (1994), no. 1-3, 225--236. MathSciNet.
[O] S. Onn, A colorful determinantal identity, a conjecture of Rota, and Latin squares. Amer. Math. Monthly 104 (1997), no. 2, 156--159. MathSciNet.
[W] M. Wild, On Rota's problem about bases in a rank matroid. Adv. Math. 108 (1994), no. 2, 336--345. MathSciNet.
[Z] P. Zappa, The Cayley determinant of the determinant tensor and the Alon-Tarsi Conjecture, Adv. Appl. Math. 19 (1997), 31--44.
* indicates original appearance(s) of problem.