On graphs decomposable into induced matchings of linear sizes

We call a graph $G$ an $(r,t)$-Ruzsa-Szemer'edi graph if its edge set can be partitioned into $t$ edge-disjoint induced matchings, each of size $r$. These graphs were introduced in 1978 and has been extensively studied since then. In this paper, we consider the case when $r=cn$. For $c>1\mathrm{/}4$, we determine the maximum possible $t$ which is a constant depending only on $c$. On the other hand, when $c=1\mathrm{/}4$, there could be as many as $\mathrm{\Omega }(\log n)$ induced matchings. We prove that this bound is tight up to a constant factor. Finally, when $c$ is fixed strictly between $1\mathrm{/}5$ and $1\mathrm{/}4$, we give a short proof that the number $t$ of induced matchings is $O(n\mathrm{/}\log n)$. We are also able to further improve the upper bound to $o(n\mathrm{/}\log n)$ for fixed $c>1\mathrm{/}4-b$ for some positive constant $b$.