There is a well-known fact that if $F$ is a family of $r$-subsets of an $n$-set no two of which intersect in exactly $s$ elements then $\vert F \vert \leq n^{\max\{s, r-s-1\}}$.
But are there any estimates if $F$ contains $r$-subsets and ANY of which intersect in MORE than $s$ elements? Should be connected to Erdos-Ko-Rado, but I can not find exactly what I am looking for.
Actually, I managed to find an answer myself.
Let's start with some definitions. $f(n, k, t) = \max \{ q = \vert F \vert\ : \forall i, j =1, 2, \dots, q \; \vert F_i \cap F_j\vert \geq t \}$.
So, the following theorem holds.
Theorem (Erdos-Ko-Rado): If $n < 2k$ then $f(n, k, 1) = {n\choose k}$, else $f(n, k, 1) = {n-1 \choose k-1}$.
Also, there is some generalization of this result.
Theorem (Ahlswede-Khachatrian): Assume that $(k−t+1)(2+ \frac{t−1}{r-1}) \leq n < (k−t+1)(2+ \frac{t−1}{r})$. Let $A_{t,r} = \{1,...,t+2r\}$. In that case $f(n,k,t)=\vert F_r \vert$, where $F_r = \{F ⊂ R_n: \vert F \vert = k, \; \vert F ∩ A_{t,r} \vert \geq t+r\}$.