I offer $1 for a proof of the following conjecture of mine, and $0.50 for a counter-example.
Let be a set with elements. Let and be partitions of into parts, each having elements.
A subset of is called a "set of representatives" for a partition if each part of contains exactly one representative.
I conjecture that there is a subset of elements of which is a set of representatives both for and for .
Canadian or US dollars?
This is known as König's theorem, and seems to be a consequence of Hall's marriage theorem. Your problem is stated in the very first paragraph of this article (but you need to pay for the article (more than $1)...).
You can also find (for free) the statement of the version generalizing both König's theorem and the marriage theorem here or there.
To Jose27: you didn't prove that two different indices i lead to different indices j, in other words that no part of P2 contains none or more than one representative.