Absolutely avoidable order-size pairs for induced subgraphs

We call a pair (m, f) of integers,m >= 1, 0 <= f <=(m/2),absolutely avoidable if there isn0such that for any pair of integers (n, e)withn>n0and 0 <= e <=(n2)there is a graph onnvertices andeedgesthat contains no induced subgraph onmvertices andfedges.Some pairs are clearly not absolutely avoidable, for example (m,0)is not absolutely avoidable since any sufficiently sparse graph onat leastmvertices contains independent sets onmvertices. Herewe show that there are infinitely many absolutely avoidable pairs.We give a specific infinite setMsuch that for anym is an element of M,thepair (m,(m2)/2) is absolutely avoidable. In addition, among otherresults, we show that for any integer functionq(m)forwhichthelimit limm ->infinity q(m)mexists, there are infinitely many values ofmsuchthat the pair (m,(m2)/2+q(m)) is absolutely avoidable