Repeated handshake counts at a 26-person party

There are $26$ people at a party, one of whom is a late arrival who shakes hands with each of the other $25$. Other handshakes happen arbitrarily, with no one shaking the same person's hand twice. Must there be two people who shook exactly the same number of hands?

Yes. Everyone has shaken at least one hand (the late arrival's) and at most $25$, so the counts lie in $\{1,\dots,25\}$. There are only $25$ possible values for $26$ people, so by pigeonhole two counts coincide.