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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.

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    I don't think that anyone on math.SE is going to be able to tell you which theorems your instructor would allow you to use, but if you are permitted to use the quoted theorem, then there is nothing left to do. If not, then you first need to say how you define the term "projective plane of order $m$". I believe that one first ought to prove that the notion of order makes sense, that is, that every line contains $m+1$ points. You might look at [this post](http://math.stackexchange.com/questions/82263/proof-of-the-number-of-nodes-in-a-finite-projective-plane/99116#99116). – 2017-01-13
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    I hadn't zeroed in on the fact that you have been asked to prove a one-way implication, not an equivalence. I think that it is likely that what you are meant to do is to show that if you take points of the 2-design to be points of a projective plane and blocks of the 2-design to be lines of the projective plane then the three axioms of a projective plane are satisfied. So you'll have to show that any two points are incident on exactly one line, that any two lines are incident on exactly one point, and that a quadrangle exists. – 2017-01-14

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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.