Three short items:

### Progress on Rota’s conjecture (polymath12) by Bucić, Kwan, Pokrovskiy, and Sudakov

First, there is a remarkable development on Rota’s basis conjecture (Polymath12) described in the paper

Halfway to Rota’s basis conjecture, by Matija Bucić, Matthew Kwan, Alexey Pokrovskiy, and Benny Sudakov

**Abstract:** In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_{i}$ (we call such bases transversal bases). Rota’s basis conjecture remains wide open despite its apparent simplicity and the efforts of many researchers (for example, the conjecture was recently the subject of the collaborative “Polymath” project). In this paper we prove that one can always find $\left(1/2-o\left(1\right)\right)n$ disjoint transversal bases, improving on the previous best bound of $\Omega\left(n/\log n\right)$. Our results also apply to the more general setting of matroids.

http://front.math.ucdavis.edu/1810.07462

Earlier the best result was giving disjoint transversal bases.

**H**ere is a subsequent paper about the more general Kahn’s conjecture

https://arxiv.org/abs/1810.07464

### Polymath 16 is alive and kicking

Polymath 16 of the chromatic number of the plane is in its eleventh post. A lot of interesting developments and ideas in various directions!

### The polymath picture

I took some pictures which are a little similar to our logo picture (last picture below). (more…)