The extremal number 
of a graph 
is the largest number of edges
in an 
-vertex
simple graph containing no subgraph
isomorphic to 
. The forbidden subgraph
need not be induced. For example, Mantel's theorem
gives 
, where 
is the cycle graph of length
three and 
is the floor function.
The example with 
has quadratic growth. More generally, the rational
exponents conjecture concerns which power-law growth rates can occur as the forbidden
graph varies.
Yang (2026) proved that the number of labeled square-free graphs on 
vertices is 
, where 
denotes little-O
notation.
-Free
Graphs." 29 Oct 2026.