HMMT November 2023 H3

There are 17 people at a party, and each has a reputation that is either $1,2,3,4$, or 5 . Some of them split into pairs under the condition that within each pair, the two people's reputations differ by at most 1. Compute the largest value of $k$ such that no matter what the reputations of these people are, they are able to form $k$ pairs.