So I have this problem:
Let there exist a 2-design with parameters $(m^2+m+1,m+1,1)$.
Prove that there exist a finite projective plane of order $m$ as well.
I have found that there is a theorem in my lectures that says:
Existence of 2-design($(m^2+m+1,m+1,1)$) $\iff $ existence of a finite projective plane of order m $\iff$ existence of a set of mutually orthogonal latin squares of order $m$.
So we didn't prove this theorem if I remember correctly because of time it would take.
Question: So can I use this theorem or should I first find a proof for the theorem?
So I know that the equivalence between latin squares and projective plane is Bose's theorem, but I don't have the proof for that first equivalence.
You cannot use this theorem; at least, if I were teaching the course I would not accept it. This is typical in higher level courses, you present a major result in class that you don't have time to prove; then as an exercise, you assign students to prove a weaker version of the result so they can see how someone might go about proving the more complete version.
It is likely you will have to prove the result yourself, and not rely on a theorem that gives it to you. To have a projective plane, you need "points" and "lines" which satisfy certain conditions. What you have is a design, containing "varieties" and "blocks" meeting their own conditions. Try to find a way to interpret the objects of your design as points/lines, and show they satisfy the projective plane properties.