I believe that there should exist theorems about finite sets which are not provable without the notion of infinite sets.
I am curious if I am right. What are the examples of such theorems if they ?
The question may be a duplicate of an early question, but I could not find one. I agree this question is vague and does not have a unique answer. Though any useful comment or answer would be appreciated.
P.S. As a possible example I guess one can think of theorems proved by mathematical induction. But I am not sure if it is relevant or not here.
A precise version of your question might be: are there statements in Peano Arithmetic (PA) that can't be proved in PA, but can be proved in set theory, say the Zermelo-Fraenkel axioms including Choice (ZFC)?
If you accept this as equivalent to your question, then there are lots of examples. The granddaddy of them all is the statement that PA is consistent. By Gödel's Second Incompleteness Theorem, this cannot be proven in PA, but it is almost trivially provable in ZFC.
As Aphelli commented, Goodstein's theorem is another famous example. So is the Paris-Harrington Principle (discussed here). That's particularly interesting, because the PHP is a slightly modified version of Ramsey's Theorem for finite sets, a famous combinatorial result. Ramsey's Theorem is provable in PA, but the PHP is not.
In other words, in ZFC we have two nearly identical proofs of two very similar statements: the infinite Ramsey theorem, and the infinite Paris-Harrington Principle. Then using compactness, we can use these to prove the finite versions. But within PA, we can prove only the finite Ramsey theorem, not the finite PHP.
It's worth noting that PA can be reformulated as a theory of finite sets. You'll find an extensive discussion of this here. Briefly, replacing the Infinity Axiom with its negation, we obtain a theory (call it ZF$\neg\infty$) that talks only about finite sets---in other words, a very good theory in which to do finite combinatorics. And there are ways to convert proofs "back and forth" between PA and ZF$\neg\infty$, so that any theorem of PA corresponds to a theorem of ZF$\neg\infty$, and vice versa.
You will notice I immediately started talking about theories, and not "notions". That's because the question is about provability. What's provable about a given notion (or mathematical concept) depends on what we assume about it. In other words, the axioms we accept.