Theorem 4 (Frankl-Wilson Theorem). Assuming that:

• $p$ is prime

• ${\lambda }_1,\dots ,{\lambda }_s$ ($s\le r$)

• ${\lambda }_i⁄\equiv r(modp)$ for each $i$

• $A\subset X^{(r)}$ such that for all distinct $x,y\in A$ have $|x\cap y|\equiv {\lambda }_i(modp)$ for some $i$