Degree of vertex in left part of a bipartite graph with distance less than 3 in right part. A friend and I have worked on the following problems and would appreciate your help generalizing the answer for $n\ge5$:
Suppose participants $P$ at a conference speak one of more of the langages in the set $L$ and that each pair can communicate in at least one language. We already proved that if $|L|=3$ and $|P| \ge10$ then one language has to be spoken by at least $2/3$ of participants. That ratio is $3/5$ if $|L|=4$ and $|P|$ is large enough to dodge low participants irregularity.
We generalize easily to the following problem and would appreciate your take :
Let $P$ and $L$ be the two sides of a bipartite graph such that for any pair $(x,y)$ of distinct vertices in $P$ then $dist(x,y)=2$. What is the lowest value for
$$p(n)=\max_{\ell\in L} \frac{\deg(\ell)}{n}$$
where $|L|=n$.

Here's an incomplete answer.
If there are $m$ participants, and at most $pm$ speak each language, then each language allows conversation between $\binom{pm}{2}$ pairs of people; there are $\binom{m}{2}$ pairs of people total, so we must have $$n \cdot \binom{pm}{2} \ge \binom{m}{2}.$$ This can only hold for arbitrarily large $m$ if $n \cdot \frac{p^2}{2} \ge \frac12$, or $p(n) \ge \frac{1}{\sqrt n}$.
To show that this is not a terrible lower bound: we have a nearly-matching upper bound when $n = q^2 + q + 1$ for some prime power $q$. In this case, there is a construction with $n$ languages and $n$ participants in which each participant speaks $q+1$ languages: just let $P$ be the set of points and $L$ be the set of lines of the projective plane of order $q$, and suppose a person speaks a language if the corresponding point lies on the corresponding line.
We can reproduce this exactly for $2n, 3n, 4n, \dots$ participants by having groups of size $2, 3, 4,\dots$ speak exactly the same set of languages, and approximately by dividing people into groups as evenly as possible. In this way, no language is spoken by more than $p(q^2+q+1) = \frac{q+1}{q^2+q+1}$ of the participants, which is a $\frac1{\sqrt n} + O(\frac1n)$ fraction. 
I expect the projective plane construction to be the best possible when it applies, but I don't know what the optimal answer is for, e.g., $p(5)$.
