Proof Writing & Problem Solving · Easy problem

The famous false "proof" that all horses have the same colour argues: given $k + 1$ horses, horses $1$ to $k$ share a colour and horses $2$ to $k + 1$ share a colour, and the two groups overlap, so all $k + 1$ do. The step "$P(k) \Rightarrow P(k+1)$" is valid for every $k$ except one. Which $k$?