MathDB

In a scientific conference, all participants can speak in total

2 \cdot n

languages (

n \geq 2

). Each participant can speak exactly two languages and each pair of two participants can have at most one common language. It is known that for every integer

k

1 \leq k \leq n-1

there are at most

k-1

languages such that each of these languages is spoken by at most

k

participants. Show that we can choose a group from

2 \cdot n

participants which in total can speak

2 \cdot n

languages and each language is spoken by exactly two participants from this group.

Problem Statement