1. Let set $A=\{0,1,2, \cdots, 9\},\left\{B_{1}, B_{2}, \cdots, B_{k}\right\}$ be a collection of non-empty subsets of $A$, and when $i \neq j$, $B_{i} \cap B_{j}$ has at most two elements. Then the maximum value of $k$ is $\qquad$
(1999, National High School Mathematics League Guangxi Preliminary Contest (High) Three))
(It is easy to see that the family of all subsets of $A$ containing at most three elements meets the requirements of the problem, where the number of subsets is $\mathrm{C}_{10}^{\mathrm{l}}+$ $\mathrm{C}_{10}^{2}+\mathrm{C}_{10}^{3}=175$. It remains to prove that this is the maximum value.)