Let $m,n$ be integers, and let $A = \{\{a,b\} \mid 1 \leq a < b\leq n\}$, the set of unordered pairs. Suppose we try to partition $A$ into subsets $A_1,\ldots,A_k$ such that each $A_i$ consists only of disjoint pairs (in other words, if $\{a,b\} \in A_i$ and $\{a,c\} \in A_i$ then $b = c$). Furthermore suppose we have the restriction that $|A_i| \leq m$ for each $i$.
Clearly a lower bound for $k$ is given by $$ k \geq \left\lceil \frac{n(n-1)}{2m} \right\rceil. $$ Is this lower bound tight in all cases? That is, can we always give a partition which allows this lower bound? If not, which tight lower bound does work?
For example, this trivially holds if $m = 1$, and it somewhat less trivially holds if $n$ is even and $m = \frac n2$.