Kovács had a dinner party for four couples. After the introductions, Mr. Kovács noted that apart from himself, each of the other nine people present had shaken hands with a different number of people.

How many people did Mrs. Kovács introduce herself to?
In the guesthouse, each of the 10 people present knew their spouse, so they did not introduce themselves to them, nor to themselves. Thus, everyone could form a maximum of 8 new acquaintances. Since, according to the problem, the 10 people present formed 9 different numbers of acquaintances, there was someone who formed 8, 7, 6, 5, 4, 3, 2, 1, 0 acquaintances. The number of handshakes made by Kovács must have matched one of these, as he was also subject to what we stated in the first sentence.

The spouse of the person who formed 8 acquaintances was the one who formed 0, since the others definitely got to know them.

Similarly, we can see step by step that the spouses of those who formed 7 and 1, 6 and 2, and 5 and 3 acquaintances are paired. For example, the spouse of the person who formed 7 acquaintances, apart from themselves and the person with $x$ acquaintances, did not introduce themselves to the person who formed 0 acquaintances, so they had to get to know all 7 of the remaining people. Thus, apart from the pairs $8-0$ and $7-x$, everyone got to know at least two people. Therefore, $x$ can only be equal to 1.

The remaining ninth person present, Kovácsné, could only form four acquaintances.