Definition 1. An equation $f(x,y,z,\dots)=0$ is partition regular (or simply regular) if, and only if, for any finite partition of $\mathbb{N}$ into $C_1, C_1,\dots, C_r$, there exists some $C_i$ containing a non-trivial solution to the equation.

Schur 最早研究了分割正规的Schur方程，也就是最简单的线性齐次方程 $x+y=z$。Schur定理表明，对自然数的任意有限染色，上述方程都有单色解。为了更好的理解，我们考虑二染色，用反证法可以证明在二染色情况下，总存在单色解。

Definition 2. Given an equation or a system of equations $\mathcal{L}$ and finite $r$ colors, the Rado number $R(\mathcal{L},r)$ is the least number such that any $r$-coloring of $\{1,2,\dots,R(\mathcal{L},r)\}$ must contain a monochromatic solution to $\mathcal{ L}$.

Theorem 3 (Rado's Theorem [1]). A linear homogeneous equation $a_1 x_1+\cdots+a_nx_n=0$ is partition regular on $\mathbb{N}$ if and only if there exists a nonempty set $J\subseteq\{1,\dots,n\}$ such that $\sum_{j\in J}a_j=0$.

Definition 4. The degree of regularity of an equation or a system of equations $\mathcal{L}$ is the largest integer $r\geq 0$, if any, such that $\mathcal{L}$ is $r$-regular. This number is denoted by $\mathrm{dor}(\mathcal{L})$. In particular, if $\mathrm{dor}(\mathcal{L})=\infty$, then $\mathcal{L}$ is regular.

$R(x^2+y^2=z^2,2)=7825$

$R(x_1,\dots,x_n,y_1,\dots,y_k)=c_1x_1+\cdots+c_nx_n+P(y_1,\dots,y_k)$

（n≥2，P是没有常数项的整系数多项式）在N上是正规的当且仅当存在非空子集 $J\subseteq \{1,\dots,n\}$ 使得 $\sum_{j\in J}c_j=0$。不仅仅如此，诸如 $x_1x_2=y_1y_2+z^2$ 等大量方程也能被证明是正规的。

$R(x+y=z^2,2)=32$

$C_R=\{1,4,5,6,7\};\\ C_B=\{2,3,8,9,10,11,12,15,18,19,20,21,24,29,30,31\}.$

$R(x+y=2z^2,2)=93.$

n=94; # then 93
# get the number of clauses for any n<100.
def clauses(n,k=0):
for j in range(2,10):
for i in range(1,j**2+1):
if i <n and 2*j**2-i<n:
k=k+2
return k

# print the head of the cpf file.
print ("p cnf",n-1,clauses(n))

# print contents of the cpf file
for j in range(2,10): # for n<100, 1 < z <10
for i in range(1,j**2+1): # get rid of symmetric solutions of x and y's
if i <n and 2*j**2-i<n: # all solutions with x, y less than n
print (i, 2*j**2-i, j, 0) # the condition of no blue monochromatic
print (-i, -2*j**2+i, -j, 0) # the condition of no red monochromatic


References.

[1] R. Rado, Studien zur Kombinatorik, Math. Z. 36 (1933), 242–280.
[2] S. W. Golomb and L. D. Baumert. "Backtrack programming." Journal of the ACM (JACM) 12.4 (1965): 516-524.
[3] M. Heule, Schur number five. Thirty-Second AAAI Conference on Artificial Intelligence. 2018.
[4] M. Heule, O. Kullmann, and V. Marek, Solving and verifying the boolean Pythagorean triples problem via cube-and-conquer. International Conference on Theory and Applications of Satisfiability Testing. Springer, Cham, 2016.
[5] M. Nasso and L. Baglini. Ramsey properties of nonlinear Diophantine equations. Advances in Mathematics 324 (2018): 84-117.
[6] B. Green and S. Lindqvist. Monochromatic Solutions to x+y=z2. Canadian Journal of Mathematics 71.3 (2019): 579-605.