I am reading “An $ n^{\frac{5}{2}}$ Algorithm for Maximum Matchings in Bipartite Graphs” by Hopcroft and Karp.

Please see the image below.

Let $ s$ be the cardinality of a maximum matching.

I think any of $ M_0, M_1, M_2, \cdots, M_s$ is a matching and $ M_s$ is a maximum matching.

So, I think $ P_s$ doesn’t exist.

But the authors wrote $ M_0, M_1, M_2, \cdots, M_i, \cdots$ and $ |P_0|, |P_1|, \cdots, |P_i|, \cdots$ .

Why?

Maybe I am confused.