On Erdos-Kakutani like Equivalents of the Failure of Continuum Hypothesis Among all mysterious equivalents of the Continuum Hypothesis and its negation, there is an algebraic combinatorial equivalent of the $\neg CH$ by Erdos and Kakutani (MR0008136) as follows:
Definition. A linear homogeneous equation $a_1x_1 + a_2x_2 + … + a_nx_n = 0$ 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. (Erdos - 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 (MR2360680) has proved the following generalization of Erdos - Kakutani theorem:

Theorem 2. (Fox) The followings are equivalent:
(a) $2^{\aleph_0}>\aleph_n$
(b) The equation $x_1+nx_2 - x_3 - ... - x_{n+3}=0$ is $\aleph_0$-regular.

My questions are in two different directions. First, note that there are proper class many instances of the failure of $CH$ in the form of $2^{\aleph_0}>\aleph_\alpha$ while the number of linear homogeneous equations $a_1x_1 + a_2x_2 + … + a_nx_n = 0$ 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 Erdos-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 $a_1x_1 + a_2x_2 + … + a_nx_n = 0$ with real coefficients such that:
$$2^{\aleph_0}>\aleph_\alpha\Leftrightarrow a_1x_1 + a_2x_2 + … + a_nx_n = 0 ~ \text{is}~\aleph_0 \text{-regular}$$

Fox and Erdos-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 Erdos-Kakutani's theorem? A non-linear equation $p(x_1,...x_n)=0$ whose $\aleph_0$-regularity is equivalent to a statement about the size of the continuum?

 A: During a personal communication, Schmerl has suggested the results in the following papers which are closely related to the answer to question 2. I add them here for the sake of those who are interested: 
Ref: 
(1) James H. Schmerl, Avoidable algebraic subsets of Euclidean space, Trans.
Amer. Math. Soc. 352 (2000), 2479-2489.
(2) James H. Schmerl, Chromatic numbers of algebraic hypergraphs, Combinatorica 37 (2017), no. 5, 1011–1026. 
(3) James H. Schmerl, Deciding the Chromatic Numbers of Algebraic Hypergrahs, (2016).

Here are some definitions and results from the above papers. You may find further information in the mentioned references:  
Definition. A polynomial $p(x_0, x_1,\cdots , x_{k-1})$ over the reals $\mathbb{R}$ is $(k, n)$-ary if each $x_i$ is an $n$-tuple of variables. We say that a $(k, n)$-ary polynomial $p(x_0, x_1,\cdots , x_{k-1})$ is avoidable if the points of $\mathbb{R}^n$ can be colored with countably many colors such that whenever $a_0, a_1,\cdots , a_{k-1}\in \mathbb{R}^n$ are distinct and $p(a_0, a_1,\cdots , a_{k-1})=0$, then there are $i < j < k$ such that the points $a_i, a_j$ are differently colored. The polynomial is unavoidable if it is not avoidable. 
Similarly, for an infinite cardinal $\kappa$, the $(k, n)$-ary polynomial  $p(x_0, x_1,\cdots , x_{k-1})$ is $\kappa$-avoidable
if the points of $\mathbb{R}^n$ can be colored using $\kappa$ colors such that whenever $a_0, a_1,\cdots , a_{k-1}\in \mathbb{R}^n$ are distinct and $p(a_0, a_1,\cdots , a_{k-1})=0$, then there are $i < j < k$ such that the points $a_i, a_j$ are differently colored. A polynomial is $\kappa$-unavoidable if it is not $\kappa$-avoidable.
In fact, $p(x_0, x_1,\cdots , x_{k-1})$ is avoidable if the chromatic number $\chi (H)$ of its zero hypergraph $H$ is countable, and it is $\kappa$-avoidable if $\chi (H)\leq\kappa$.
Example 1. For each $k<\omega$ and ordinal $\alpha$, the polynomial, $x_0+\cdots+x_{k}-x_{k+1}-kx_{k+2}$, in Fox' theorem is a $(k + 3, 1)$-ary polynomial
which is $\aleph_{\alpha}$-avoidable if and only if $2^{\aleph_0}\leq \aleph_{\alpha+k}$.
Example 2. The $(3, n)$-ary polynomial $p(x, y, z)=||x-y||^2-||y-z||^2$, for $2\leq n<\omega$ where $||.||$ denotes the Euclidean norm in $\mathbb{R}^n$, is avoidable. 
Theorem 1. For each $n < w$ there is a countable partition of $\mathbb{R}^n$ which avoids every avoidable $(-, n)$-ary polynomial over $\mathbb{Q}$. 
Theorem 2. Assuming $2^{\aleph_0}\geq \aleph_{m}$, every avoidable polynomial is $m$-avoidable.  
Theorem 3.  Assuming $2^{\aleph_0}\leq \aleph_{m}$, every $m$-avoidable polynomial is avoidable.
Corollary. The followings hold:
(a) Assuming $2^{\aleph_0}\geq \aleph_{m}$, a polynomial is avoidable iff it is $m$-avoidable. 
(b) Assuming $2^{\aleph_0}\leq \aleph_{m}$, a polynomial is avoidable iff it is $\omega$-avoidable.  
