The induced rational exponents conjecture holds for r=2-a/b where b ≥ max{a, (a-1)^2}.
Erd˝ os, On the combinatorial problems which I would most like to see solved,Combinatorica 1(1981), 25-42
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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}.