  The minrank of a graph $G$ is the minimum rank of a matrix $M$ that can be
obtained from the adjacency matrix of $G$ by switching some ones to zeros
(i.e., deleting edges) and then setting all diagonal entries to one. This
quantity is closely related to the fundamental information-theoretic problems
of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and
distributed storage, and to Valiant's approach for proving superlinear circuit
lower bounds (Valiant, Boolean Function Complexity '92).
  We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs
$G(n,p)$ for all regimes of $p\in[0,1]$. In particular, for any constant $p$,
we show that $\mathsf{minrk}(G) = \Theta(n/\log n)$ with high probability,
where $G$ is chosen from $G(n,p)$. This bound gives a near quadratic
improvement over the previous best lower bound of $\Omega(\sqrt{n})$ (Haviv and
Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky
and Stav (FOCS '07). Our lower bound matches the well-known upper bound
obtained by the "clique covering" solution, and settles the linear index coding
problem for random graphs.
  Finally, our result suggests a new avenue of attack, via derandomization, on
Valiant's approach for proving superlinear lower bounds for logarithmic-depth
semilinear circuits.
