The Janson-Sós conjecture characterizes a quasirandom graph sequence by balanced subgraph counts. Fix a finite simple
graph 
with 
vertices and 
edges, and fix 
. For disjoint ordered vertex
sets 
in 
,
let 
be the average, over all labelings of 
, of its injective copies with its 
th labeled graph vertex in
.
Copies need not be induced subgraphs.
The conjecture asserts that
 |
uniformly over all such choices with 
implies that 
is 
-quasirandom, where 
and 
is the floor function.
The classes may leave fewer than 
vertices unused. Wang et
al. (2026) proved this assertion, including when 
has isolated vertices.
The restriction 
is necessary, since the assertion fails for a single graph
edge with possible isolated vertices when
.
