rainbow_base_cover/notes/main.md

title

title
Matroid Rainbow Base Cover

Let M=(E,\mathcal B) be a rank-r matroid whose ground set decomposes into two disjoint bases. Furthermore, assume that E is colored by r colors, each color appearing exactly twice. A basis of M is called rainbow if it does not contain two elements of the same color.

::: Problem What is the minimum number of rainbow bases needed to cover E? :::

Kristóf Bérczi conjectured the minimum number is 3.

Currently known bounds:

• upperbound: \floor{\log_2 |E|}+1 by matroid intersection;
• lowerbound: 3 on graphic matroid of K_4.

::: {.Conjecture #conj-doublecover} Let M_1 and M_2 be two matroid on the same ground set E. Assume that E decomposes into two bases in both of them. Then M_1 and M_2 has four common bases that cover each element exactly twice. :::

Let M_1 be the partition matroid of colors and let M_2 be M. If [@conj-doublecover] is true, then 3 common bases will be enough to cover E.