Laci and Kati
This is the first of a few posts which are spin-offs of the extremal combinatorics series, especially of part III. Here we talk about Lovasz’s geometric two families theorem.
1. Lovasz’s two families theorem
Here is a very beautiful generalization of the two families theorem due to Lovasz. (You can find it in his 1977 paper Flats in Matroids and Geometric Graphs.)
Lovasz’s Theorem: Let and be two families of linear spaces having the following properties:
1) for every , , and .
2) For every , ,
3) for every , .
This theorem is interesting even if all vector spaces are subspaces of an -dimensional space.
Recall Bollobas’s two families theorem: Continue reading