  We call a pair $(m,f)$ of integers, $m\geq 1$, $0\leq f \leq \binom{m}{2}$,
\emph{absolutely avoidable} if there is $n_0$ such that for any pair of
integers $(n,e)$ with $n>n_0$ and $0\leq e\leq \binom{n}{2}$ there is a graph
on $n$ vertices and $e$ edges that contains no induced subgraph on $m$ vertices
and $f$ edges. Some pairs are clearly not absolutely avoidable, for example
$(m,0)$ is not absolutely avoidable since any sufficiently sparse graph on at
least $m$ vertices contains independent sets on $m$ vertices. Here we show that
there are infinitely many absolutely avoidable pairs. We give a specific
infinite set $M$ such that for any $m\in M$, the pair $(m, \binom{m}{2}/2)$ is
absolutely avoidable. In addition, among other results, we show that for any
monotone integer function $q(m)$, $|q(m)|=O(m)$, there are infinitely many
values of $m$ such that the pair $(m, \binom{m}{2}/2 +q(m))$ is absolutely
avoidable.
