# Choice functions in the intersection of matroids

## TL;DR

This paper generalizes existing results on rainbow matchings and matroid intersections, demonstrating the existence of a rainbow set supporting a function with specified independence properties across multiple matroids.

## Contribution

It introduces a unified framework that extends previous theorems on rainbow fractional matchings and matroid intersections to a broader setting.

## Key findings

- Established a common generalization of two key results in matroid theory.
- Proved the existence of a rainbow set supporting a function with independence in multiple matroids.
- Extended the applicability of rainbow set theorems to more complex matroid configurations.

## Abstract

We prove a common generalization of two results, one on rainbow fractional matchings and one on rainbow sets in the intersection of two matroids: Given $d = r \lceil k \rceil - r + 1$ functions of size (=sum of values) $k$ that are all independent in each of $r$ given matroids, there exists a rainbow set of $supp(f_i)$, $i \leq d$, supporting a function with the same properties.