# Maximal number of subsets with 3 elements and small intersection - special Constant Weight Codes

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

1. $|Y_i|=3$ for all $i=1,\dots, k$
2. $|Y_i\cap Y_j|\leq 1$ 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.

Cheers, Sofie

I'll just adress you question as to whether there is $N$ so that $k(N)\geq 2n$. Yes, it is very possible. In fact if $N$ is $1$ or $3\bmod 6$ we can always find $\frac{n(n-1)}{6}$ subsets with this property by taking a steiner triple system.
In particular a Steiner triple system of size $13$ would have $\frac{13\cdot12}{6}=26$ subsets of size $3$. So indeed it is possible for $k(N)$ to be greater than $2n$, or $sn$ for any integer $s$ for that matter.