The coefficients $a_1, \dots, a_k$ satisfy Rado's condition if $\sum_{i \in I} a_i = 0$ for some non-empty $I \subseteq [k]$.

For a single homogeneous equation this is exactly the criterion of Rado's theorem [Ra33]: $a_1x_1 + \cdots + a_kx_k = 0$ is partition regular if and only if the coefficients satisfy it.

$c(a_1, \dots, a_k)$, the least number of colours required in order to colour $\mathbb{N}$ so that there is no monochromatic solution to $a_1x_1 + \cdots + a_kx_k = 0$.

The solutions $x_i$ are required to be positive, as in [FoKl06]: were $0$ admitted then $x_1 = \cdots = x_k = 0$ would be a monochromatic solution for every colouring, and no number of colours would ever suffice. The $x_i$ are not required to be distinct.

When the coefficients do not satisfy RadoCondition, Rado's theorem [Ra33] guarantees that some finite colouring has no monochromatic solution, so the set below is non-empty and this sInf is a genuine minimum.

Suppose that $a_1, \dots, a_k$ are integers which do not satisfy Rado's condition: thus if $\sum_{i \in I} a_i = 0$ then $I = \emptyset$. It then follows from Rado's theorem that the equation $a_1x_1 + \cdots + a_kx_k = 0$ is not partition regular. Write $c(a_1, \dots, a_k)$ for the least number of colours required in order to colour $\mathbb{N}$ so that there is no monochromatic solution to $a_1x_1 + \cdots + a_kx_k = 0$. Is $c(a_1, \dots, a_k)$ bounded in terms of $k$ only?

This problem, which is known as Rado's boundedness conjecture, dates back to 1933 [Ra33]. It is open for all $k \geq 4$.

The answer was shown to be affirmative for $k = 3$ by Fox and Kleitman [FoKl06], who showed that $c(a_1, a_2, a_3) \leq 24$.

Green [Gr24] is not sure that the constant $24$ of [FoKl06] is sharp, and remarks that it might be interesting to determine the sharp constant.

The largest value of $c(a_1, a_2, a_3)$ is attained, since by green_21.variants.fox_kleitman the values form a non-empty set of naturals bounded above by $24$.

A question [FoKl06, Conjecture 5] of Fox and Kleitman, which they call a 'modular analogue' of Rado's Boundedness Conjecture. Let $p$ be a prime, and suppose that $a_1, \dots, a_k$ are integers with $\sum_{i \in I} a_i \equiv 0 \pmod p$ only when $I = \emptyset$. Does there exist an $f(k)$-colouring of $(\mathbb{Z}/p\mathbb{Z})^*$ with no monochromatic solution to $a_1x_1 + \cdots + a_kx_k = 0$? This seems to be open even when $k = 3$; Green [Gr24] suspects the answer may be negative.

The point of the question is that the number of colours $f(k)$ must not depend on $p$.

The largest $d \leq r$ such that $\sum_{i \in I} a_i \equiv 0 \pmod{2^d}$ for some non-empty subset $I \subseteq [k]$, where $a_1, \dots, a_k \in \mathbb{Z}/2^r\mathbb{Z}$.

Congruence mod $2^d$ of an element of $\mathbb{Z}/2^r\mathbb{Z}$ is expressed through its canonical representative ZMod.val; this is unambiguous because $d$ is capped at $r$.

Milićević [ElJo23, Conjecture 11.1] conjectures the following 2-adic variant. For any $k \in \mathbb{N}$, there exists $K = K(k)$ such that the following is true. Let $r$ be a positive integer, and let $a_1, \dots, a_k \in \mathbb{Z}/2^r\mathbb{Z}$. Let $d$ be the largest integer such that $\sum_{i \in I} a_i \equiv 0 \pmod{2^d}$ for some non-empty subset $I \subset [k]$. Then there is a $K$-colouring of $\mathbb{Z}/2^r\mathbb{Z}$ such that all monochromatic solutions $x = (x_1, \dots, x_k)$ to the equation $a_1x_1 + \cdots + a_kx_k = 0$ satisfy $x_i \equiv 0 \pmod{2^{r-d}}$ for all $i = 1, \dots, k$.

Milićević remarks that, if true, this would imply the Rado boundedness conjecture by a compactness argument.

Not satisfying RadoCondition is Green's phrasing "if $\sum_{i \in I} a_i = 0$ then $I = \emptyset$".

A tuple failing RadoCondition has no zero coefficient, since a singleton is a non-empty subset.

In the degenerate case $k = 0$ the equation reads $0 = 0$, so the empty tuple is a monochromatic solution for every colouring and minColours takes its junk value 0. The bound asserted by green_21 is therefore vacuously satisfied at $k = 0$, and the content of the problem is unaffected.

A case where the infimum defining minColours is a genuine minimum rather than the junk value of sInf ∅. The coefficients $(1, 1, 1)$ fail RadoCondition, and since the $x_i$ are positive the equation $x_1 + x_2 + x_3 = 0$ has no solutions at all, so a single colour suffices; no colouring of $\mathbb{N}$ into Fin 0 exists, so $0$ is not attainable.