The polynomial van der Waerden number 
is the least positive
integer 
such that every 
-coloring of 
contains distinct equally colored 
satisfying 
for some integer 
. Here 
is a nonzero integer-coefficient
polynomial with 
. Existence follows from the polynomial extension
of van der Waerden's theorem (Bergelson
and Leibman 1996). The integer 
is not restricted to be positive, and 
must be nonzero.
Kumthekar et al. (2026) determined nine exact four-color quadratic values. For coefficient pairs 
, 
, 
, 
, 
, 
, 
, 
, and 
,
the corresponding 
values are respectively 97, 49, 33, 25, 49, 29,
57, 41, and 17. For all integer 
not both zero, they proved
 |
The optimal universal coefficient in the upper bound lies between 57 and 64. The uniform upper bound uses exact non-four-colorability computations on a finite polynomial-distance graph.
