Let $n$ be a positive integer. If $A_1,A_2,\ldots,A_m$ are $3$-subsets of $[n]$ such that $\left|A_i\cap A_j\right|\neq 1$ for $i\neq j$, then the largest possible value of $m$ is
$$m_\max=\left\{
\begin{array}{ll}
n&\text{if }n\equiv0\pmod{4}\,,\\
n-1&\text{if }n\equiv1\pmod{4}\,,\\
n-2&\text{else}\,.
\end{array}
\right.$$
Remark: Below is a sketch of my proof of the claim above. Be warned that a complete proof is quite long, whence I am providing a sketch with various gaps to be filled in. I hope that somebody will come up with a nicer proof.
Proof. The first two cases follow from my first answer. I shall now deal with the last case, where $m_\max=n-2$.
Suppose contrary that there are $A_1,A_2,\ldots,A_{n-1}$ satisfying the intersection condition. Then, proceed as before. The indicator vectors $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{n-1}\in\mathbb{F}_2^n$ are linearly independent. Thus, there exists $\mathbf{v}\in\mathbb{F}_2^n$ such that $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{n-1},\mathbf{b}$ form a basis of $\mathbb{F}_2^n$. We can assume that $\langle \mathbf{a}_i,\mathbf{b}\rangle=0$ for all $i=1,2,\ldots,n-1$ (otherwise, replace $\mathbf{b}$ by $\mathbf{b}-\sum_{i=1}^{n-1}\,\langle \mathbf{a}_i,\mathbf{b}\rangle \,\mathbf{a}_i$). Observe that $\langle \mathbf{b},\mathbf{b}\rangle=1$.
Note that $$\boldsymbol{1}=\sum_{i=1}^{n-1}\,\mathbf{a}_i+\mathbf{b}\,.$$
Let $B$ be the subset of $[n]$ with the indicator vector $\mathbf{b}$. Let $X$ denote the set of $i$ such that $A_i$ is disjoint from $B$, and $Y$ the set of $i$ such that $A_i\cap B$ has two elements. Observe that $X$ and $Y$ form a partition of $\{1,2,\ldots,n-1\}$; moreover,
$$\mathcal{X}:=\bigcup_{i\in X}\,A_i\text{ and }\mathcal{Y}:=\bigcup_{i\in Y}\,A_i$$
are disjoint subsets of $[n]$.
If $X\neq \emptyset$, then we can use induction to finish the proof, noting that $A_i\subseteq [n]\setminus (B\cup\mathcal{Y})$ for all $i\in X$. From now on, assume that $X=\emptyset$.
Consider a simple graph $G$ on the vertex set $B$ where two vertices $i,j\in B$ ($i\neq j$) are connected by an edge iff $i$ and $j$ belongs in some $A_p$ simultaneously. If $C$ is a connected component of $G$ and $k\in [n]\setminus B$, then we say that $k$ is adjacent to $C$ if there exists $A_p$ such that $A_p\cap B$ is an edge of $C$ and $k\in A_p$, in which case, we also say that $A_p$ is incident to $C$. It is important to note that, if $C_1$ and $C_2$ are two distinct connected components of $G$, and $k_1,k_2\in [n]\setminus B$ are adjacent to $C_1$ and $C_2$, respectively, then $k_1\neq k_2$.
Let $C$ be a connected component of $G$ with at least two vertices. We have three probable scenarios:
- $C$ is a type-1 connected component, namely, $C$ is an isolated edge (i.e., it has only two vertices and one edge);
- $C$ is a type-2 connected component, namely, $C$ is a triangle (i.e., $C$ consists of $3$ vertices and $3$ edges);
- $C$ is a type-3 connected component, namely, $C$ is a star graph (i.e., there exists a vertex $v$ of $C$ such that every edge of $C$ takes the form $\{v,w\}$, where $w$ is any vertex of $C$ distinct from $v$).
It can be readily seen that, if $C$ is a connected component of type 2 or type 3 of $G$, then $C$ is adjacent to exactly one element of $[n]\setminus B$. If $G$ has a connected component $C$ of type 2, then the removal of vertices in $C$ along with the element $j\in[n]\setminus B$ which is adjacent to $C$ reduces the elements of $[n]$ by $4$, whilst ridding of only three sets $A_i$. Then, we finish the proof for this case by induction. Suppose from now on that $G$ has no connected components of type 2.
Now, assume that $G$ has a connected component $C$ of type 3, which has $s$ vertices. Let $j\in[n]\setminus B$ be adjacent to $C$. Then, the removal of vertices of $C$ along with $j$ from $[n]$ reduces the elements of $[n]$ by $s+1$, whilst ridding of only $s-1$ sets $A_i$. Therefore, the claim hold trivially.
Finally, assume that $G$ has only connected components of type 1 and possibly some isolated vertices. Then, it follows immediately that there are at most $n-2t$ sets $A_i$, where $t$ is the number of connected components of type 1. This shows that $t=0$. Thus, $G$ has only isolated vertices, but this is a contradiction as well (as $X=\emptyset$ is assumed).