Question 7
There is a population $P$ of $10000$ bacteria, some of which are friends (friendship is mutual), so that each bacterion has at least one friend and if we wish to assign to each bacterion a coloured membrane so that no two friends have the same colour, then there is a way to do it with $2021$ colours, but not with $2020$ or less.
Two friends $A$ and $B$ can decide to *merge* in which case they become a single bacterion whose friends are precisely the union of friends of $A$ and $B$. (Merging is not allowed if $A$ and $B$ are not friends.) It turns out that no matter how we perform one merge or two consecutive merges, in the resulting population it would be possible to assign $2020$ colours or less so that no two friends have the same colour. Is it true that in any such population $P$ every bacterium has at least $2021$ friends?
**Proposed by Bulgaria**
Two friends $A$ and $B$ can decide to *merge* in which case they become a single bacterion whose friends are precisely the union of friends of $A$ and $B$. (Merging is not allowed if $A$ and $B$ are not friends.) It turns out that no matter how we perform one merge or two consecutive merges, in the resulting population it would be possible to assign $2020$ colours or less so that no two friends have the same colour. Is it true that in any such population $P$ every bacterium has at least $2021$ friends?
**Proposed by Bulgaria**