Question:
Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\{1,2,\ldots,1000\}$ there exist two of them which share at least $r$ elements.
By Now, I claim that $\color{red}{ r \le 200} $**
reasons were as follows
For all $ k \in \{1, 2, \dots, 10\} $ let $\color{red}{ A_k = \{100k - 99, 100k - 98, \dots, 100k\}}. $ Then if we look at the subsets $ A_1 \cup A_5 \cup A_6 \cup A_7 \cup A_9 $ and $ A_1 \cup A_2 \cup A_7 \cup A_8 \cup A_{10} $ and $ A_2 \cup A_3 \cup A_6 \cup A_8 \cup A_{9} $ and $ A_3 \cup A_4 \cup A_7 \cup A_9 \cup A_{10} $ and $ A_4 \cup A_5 \cup A_6 \cup A_8 \cup A_{10} $ we see that any two of these subsets share exactly $ 200 $ elements which implies that $\color{blue}{ r \le 200. }$
I conjecture $\color{red}{r_{\max}=200?}$ and I can't prove it.Thanks