Among all mysterious equivalents of the Continuum Hypothesis and its negation, there is an algebraic combinatorial equivalent of $\neg \mathit{CH}$ in Erdös and Kakutani - On non-denumerable graphs (MR0008136) as follows:
Definition. A linear homogeneous equation $a_1x_1 + a_2x_2 + \dotsb + a_nx_n = 0$ with real coefficients is called $\aleph_0$-regular if every coloring of the real numbers by $\aleph_0$-many colors has a monochromatic solution to the above equation in distinct $x_i$.
Remark. Not all linear homogeneous equations with real coefficients are $\aleph_0$-regular. An example is Schur's equation: $x_1 + x_2-x_3=0$.
Theorem 1. (Erdös – Kakutani) The followings are equivalent:
(a) $2^{\aleph_0}>\aleph_1$.
(b) The equation $x_1+x_2 - x_3 - x_4=0$ is $\aleph_0$-regular.
In the other words, not only there must be many reals if one can always find a monochromatic solution for such an equation with respect to every coloring, but also if there are so many reals then such a solution actually exists for every $\aleph_0$-coloring.
Inspired by some results of Komjath, Fox - An infinite color analogue of Rado's therorem (MR2360680) proved the following generalization of the Erdös – Kakutani theorem:
Theorem 2. (Fox) The followings are equivalent:
(a) $2^{\aleph_0}>\aleph_n$.
(b) The equation $x_1+nx_2 - x_3 - \dotsb - x_{n+3}=0$ is $\aleph_0$-regular.
My questions are in two different directions. First, note that there are proper class many instances of the failure of $\mathit{CH}$ in the form of $2^{\aleph_0}>\aleph_\alpha$ while the number of linear homogeneous equations $a_1x_1 + a_2x_2 + \dotsb + a_nx_n = 0$ with real coefficients is limited. Thus there must be a place where the correlation between the size of continuum and $\aleph_0$-regularity of linear homogeneous equations with real coefficients breaks! So there must be a minimum $\alpha$ where there is NO Erdös–Kakutani-like equivalent of the $2^{\aleph_0}>\aleph_\alpha$. How far is it?
Question 1. What is the minimum ordinal $\alpha$ such that there is NO linear homogeneous equation $a_1x_1 + a_2x_2 + \dotsb + a_nx_n = 0$ with real coefficients such that:
$$2^{\aleph_0}>\aleph_\alpha\iff a_1x_1 + a_2x_2 + … + a_nx_n = 0 ~ \text{is}~\aleph_0 \text{-regular}$$
The Fox and Erdös–Kakutani theorems indicate that the existence of monochromatic solutions for certain linear equations over real numbers contains valuable information about the size of the continuum. What about arbitrary equations, particularly the simple non-linear ones? And what sort of information do they contain? Can their $\aleph_0$-regularity impose upper bounds on the size of the continuum rather than lower bounds?
Question 2. Is there any non-linear version of Erdös–Kakutani's theorem? A non-linear equation $p(x_1,\dotsc x_n)=0$ whose $\aleph_0$-regularity is equivalent to a statement about the size of the continuum?