Fraction of diners who know a seat neighbour

At a party, $2m$ people ($m>1$) sit in random order around a round table. They carry distinct labels $0,1,\dots,2m-1$, and the person labelled $i$ knows exactly $i$ of the other $2m-1$ guests. Acquaintance need not be mutual: $B$ knowing $A$ does not mean $A$ knows $B$. What is the expected fraction of guests who know at least one of the two people seated beside them?

$2/3$. Seating is random, so person $k$ knows neither neighbour with probability $\binom{2m-1-k}{2}/\binom{2m-1}{2}$. Summing over $k$ via the hockey-stick identity gives $\binom{2m}{3}$, so the average of these probabilities is exactly $\tfrac13$; by linearity the expected fraction is $\tfrac23$.