The rational exponents conjecture of Erdős and Simonovits asserts that every rational number ![r in [1,2]](/images/equations/RationalExponentsConjecture/Inline1.svg)
occurs as an exponent of the extremal
number of a single graph. More precisely, there is
a graph 
for which
 |
where big-Theta notation describes the growth as 
.
Here 
is the maximum number of edges in an 
-vertex simple graph containing
no copy of 
.
The formulation for 
is problem
571 in the Erdős problems collection.
Conlon et al. (2026) proved the conjecture for a single forbidden graph, rather than merely a finite family of forbidden graphs. Conlon et al. (2026) present a proof originating in Epoch AI's benchmarking of GPT-6 Astra and explain its dependence on earlier extremal constructions.
