Monochromatic sums and products in $\mathbb{N}$

TL;DR

The paper proves that any finite coloring of \(\mathbb N\) contains infinitely many \(x,y\in\mathbb N\) such that \(\{x,x+y,xy\}\) is monochromatic, answering the long-standing question for the pair \(\{x+y,xy\}\) in a strong form. It does so via a new affine topological correspondence principle and derives partition regularity for broader polynomial configurations and equations such as \(a^2-b^2=c\).

Key findings

• Corollary 1.5: for any finite coloring of \(\mathbb N\), there exist infinitely many \(x,y\in\mathbb N\) with \(\{x,xy,x+y\}\) monochromatic; in particular, \(\{x+y,xy\}\) is monochromatic.
• Theorem 1.4 gives a general class: for any \(s\in\mathbb N\) and finite sets \(F_i\) of functions \(\mathbb N^i\to\mathbb Z\) whose last-variable functions are polynomial with zero constant term, any finite coloring yields a color \(C\) and infinitely many \((s+1)\)-tuples \(x_0,\dots,x_s\in\mathbb N\) such that \(x_0\cdots x_s\in C\) and \(x_1\cdots x_j+f(x_{j+1},\dots,x_i)\in C\) for all \(0\le j<i\le s\) and \(f\in F_{i-j}\).
• Corollary 1.7: if \(c_1+\cdots+c_k=0\) with \(c_i\in\mathbb Z\setminus\{0\}\), then any finite coloring of \(\mathbb N\) has pairwise distinct \(a_0,\dots,a_k\) of the same color satisfying the corresponding partition-regular equation; Corollary 1.8 specializes to a monochromatic solution of \(a^2-b^2=c\).
• Theorem 7.5 extends the main combinatorial result to large ideal domains (LIDs), a class including fields, \(\mathbb Z\), rings of integers of number fields, and \(\mathbb F[x]\).

Methods

• The proof is mathematical/combinatorial, with no empirical data or sample. It first transfers the problem to topological dynamics via an affine correspondence principle (Theorem 3.2), using ultrafilters and the Stone–Čech compactification, reducing Theorem 1.4 to a dynamical statement (Theorem 3.1).
• The dynamical proof uses a “complexity reduction” technique inspired by Bergelson–Leibman, the polynomial van der Waerden theorem (Theorem 4.1/Corollary 4.2), and a multiplicative-derivative identity for expressions \(M_nA_{f(n)}\).
• Section 5 also gives a short, purely combinatorial proof of Corollary 1.5 using piecewise syndetic sets and a van der Waerden-type theorem (Theorem 5.1).

Limitations

• Stated: the paper says a complete solution to Problem 1.2—characterizing all polynomial Ramsey families—is still very far from reach; the results cover a large class, not a full classification.
• Stated: the methods of earlier work on infinite fields cannot be directly used for \(\mathbb N\), because the affine semigroup of \(\mathbb N\) is not amenable.
• Stated: the LID extension relies on Lemma 7.4, which fails in general rings, even some principal ideal domains; the paper gives \(\mathbb Q[x]\) as an example where \(x\mathbb Q[x]\) has infinite additive index and cannot be piecewise syndetic. Also, \(\mathbb N\) is not strictly a LID, and LID semirings are not pursued.