A Diophantine Ramsey Theorem
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Bolyai Society – Springer-Verlag
Combinatorica 19pp. DOI: 10.1007/s00493-020-4482-5
A DIOPHANTINE RAMSEY THEOREM TOMASZ SCHOEN Received March 6, 2020 Revised September 17, 2020 Let p ∈ Z[x] be any polynomial P with p(0) = 0, k ∈ N and let c1 , . . . , cs ∈ Z, s > k(k + 1), be non-zero integers such that ci = 0. We show that for a wide class of coefficients c1 , . . . , cs in every finite coloring N = A1 ∪· · ·∪Ar there is a monochromatic solution to the equation c1 xk1 + · · · + cs xks = p(y).
1. Introduction For a polynomial P ∈ Z[x1 , . . . , xs ] we call the equation P (x1 , . . . , xs ) = 0 regular if in any finite partition N = A1 ∪· · ·∪Ar there is a non-trivial solution to this equation with x1 , . . . , xs ∈ Ai for some 1 6 i 6 r. Throughout the course of the paper by a trivial solution we mean a solution with x1 = · · · = xs . The study of regular equations was started by Schur [26], who showed that x + y = z is regular. Later Rado [24] proved the following theorem that provides a complete characterization of regular linear equations. Theorem 1 (Rado [24]). Let c1 , . . . , cs ∈ Z\{0} and s > 3. Then the linear equation c1 x1 + · · · + cs xs = 0 P is regular if and only if there is a non-empty set I ⊆ [s] such that i∈I ci = 0. Recently, many various questions concerning regularity of equations were investigated [4], [5], [6], [7], [8], [10], [13], [17], [19], [21], [22]. Specifically, Mathematics Subject Classification (2010): 11P99; 05D10
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a number of authors attempted to find a Rado-type characterization for Diophantine equations [4], [5], [20]. The most general result was obtained by Chow, Lindqvist and Prendiville [5], who have proved a Rado criterion for k powers: if s(k) ≥ (1 + o(1))k log k then the equation (1)
c1 xk1 + · · · + cs xks = 0
P is regular if and only if there is a non-empty set I ⊆ [s] such that i∈I ci = 0 (it is proven in [5] that one can even find such solution with distinct integers). The above result despite being very close to the best possible one it does not provide full characterization of regular equations for k powers. It is clear that we need a lower bound for the number of variables in terms of k, however it seems to be a very complicated matter related to Waring’s problem, to find optimal value of s(k). A result towards longstanding open problem concerning the regularity of the Pythagorean equation x2 + y 2 = z 2 was proven in [5], specifically the equation x21 + x22 + x23 + x24 = x25 is regular. Moreira [22] showed that the equation c1 x21 + · · · + cs x2s = y P is regular provided that i ci = 0 and s > 2. Another result was obtained by Bergelson [1] using the ergodic theory method, who showed regularity of the equations (2)
x − y = p(z),
where p ∈ Z[x] is arbitrary polynomial with p(0) = 0. In contrast, Green and Lindqvist [10] observe that the equation x + y = z 2 is not 3-regular (there is a 3-coloring of N without monochromatic solutions) and they used a very elaborate argument to prove that every 2-coloring of N has a monochromatic solution t
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