The induced rational exponents conjecture holds for r=2-a/b where b ≥ max{a, (a-1)^2}.
F¨ uredi, On a Tur´ an type problem of Erd˝ os,Combinatorica11(1991), 75-79
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Induced rational exponents near two
The induced rational exponents conjecture holds for r=2-a/b where b ≥ max{a, (a-1)^2}.