Let
![\alpha < \frac {3 - \sqrt {5}}{2}](/media/m/7/9/4/794a4b3bbe02909604fd12d75322d5ef.png)
be a positive real number. Prove that there exist positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
and
![p > \alpha \cdot 2^n](/media/m/8/0/c/80cfb2d81c26534d3ee191fd7954e0c8.png)
for which one can select
![2 \cdot p](/media/m/3/8/2/38267a661ef73f749f2470e489778624.png)
pairwise distinct subsets
![S_1, \ldots, S_p, T_1, \ldots, T_p](/media/m/d/e/8/de8cfbd0f9c6052272fb9bcedb4f6cb5.png)
of the set
![\{1,2, \ldots, n\}](/media/m/8/3/2/8329a95a540f5c7a2a5ceff914234e40.png)
such that
![S_i \cap T_j \neq \emptyset](/media/m/0/e/c/0ec301be75728d3c61aca3b35f2414e2.png)
for all
Author: Gerhard Wöginger, Austria
%V0
Let $\alpha < \frac {3 - \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $n$ and $p > \alpha \cdot 2^n$ for which one can select $2 \cdot p$ pairwise distinct subsets $S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $\{1,2, \ldots, n\}$ such that $S_i \cap T_j \neq \emptyset$ for all $1 \leq i,j \leq p$
Author: Gerhard Wöginger, Austria