in the middle of some proof I encountered a combinatorial problem and tracked it back to the theory of Constant Weight Codes. Those problems seem hard to solve, but my question is rather specific, so I am still posting it in the hope for ideas:
Assume you have a set $X$ of $N$ elements (all numbers are finite). What is the largest number $k=k(N)$ such that there exist subsets $Y_1, \dots, Y_k\subseteq X$ with
- $|Y_i|=3$ for all $i=1,\dots, k$
- $|Y_i\cap Y_j|\leq 1$ for all $i\neq j$.
Ultimately I am not even interested in the explicit number, but rather in an answer to the following
Question: Does there exist some natural number $N$, such that $k(N)\geq 2N$?
In the language of Constant Weight Codes, the number $k(N)$ seems to refer to the maximal number $A(N,4,3)$ of Codewords in a binary Constant Weight Code with length $N$, Hamming distance $4$ and constant weight $3$.
I would be very thankful for suggestions.