aa r X i v : . [ m a t h . C O ] J a n A Note on a Question of Erd˝os & Graham
Kellen MyersJune 24, 2018
Abstract
Erd˝os & Graham ask whether the equation x + y = z is partitionregular, i.e. whether it has a finite Rado number. This note providesa lower bound and also states results in the affirmative for two similarquadratic equations. In [Sch16], Schur shows that for any finite coloring of the positive integers (afunction χ : Z + → { , , . . . , r } ), there will be a triple ( x, y, z ) such that x + y = z and χ ( x ) = χ ( y ) = χ ( z ). This triple is said to be a monochromatic solutionto the equation x + y = z . The least N such that this statement holds for anycoloring χ : { , , . . . , N } → { , , . . . , r } is called the r -color Schur number andis denoted S ( r ).It is easy to see that S (2) = 5, and one might note trivially that S (1) = 2. Thevalues of S (3) and S (4) are also known. We can associate such a quantity toany equation E , not just x + y = z , which denote R r ( E ). In cases where no such N exists, we say R r ( E ) = ∞ . An equation E is called r -regular if the quantity R r ( E ) is finite, and it is called regular if it is r -regular for all r .In [Rad33], Rado provides necessary and sufficient conditions for linear, homoge-neous equations to be regular. He also gives necessary and sufficient conditions(essentially non-triviality) for such equations to be 2-regular. For that reason,we call these quantities “Rado numbers.” There have been many papers, start-ing with [BB82], giving certain Rado numbers (often parametrized families ofequations). In most cases the equations are linear and there are 2 colors.In this note, we state three results from a forthcoming paper [MP] that do notfall into these categories. The equations are quadratic, and in one case, thenumber of colors is 3. 1n [EG80], Erd˝os and Graham ask whether the equation x + y = z is 2-regular.In [Gra08], Graham notes that it is not clear which answer is correct. In [FH13]it is proved that 9 x + 16 y = n (along with a family of similar quadraticequations) is 2-regular, but only x and y are supposed to be monochromatic(note n ). We offer the following three results, which have been a part of ongoingwork to settle the question: Theorem 1. R (cid:0) x + y + z = w (cid:1) = 105 . Theorem 2. R (cid:0) x + x + x + x = y + y + y (cid:1) = 32 . Theorem 3. R (cid:0) x + y = z (cid:1) > . In the third of these results, we do not exclude the possibility that it is infinite.These results are all obtained computationally and will be detailed in [MP].The first result also inspires a new sequence representing the Rado number forthe equation x + x + · · · + x k = z , which is tabulated as follows: k = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 N = ? 105 37 23 18 20 20 15 16 20 23 17 21 26 17 23This is now entry A250026 in the Online Encyclopedia of Integer Sequences. References [BB82] Albrecht Beutelspacher and Walter Brestovansky,
Generalized SchurNumbers , Lecture Notes in Mathematics (1982), 30–38.[EG80] Paul Erd˝os and Ron Graham,
Old and New Problems and Results inCombinatorial Number Theory , Universit´e de Gen`eve, L’EnseignementMath´ematique (1980).[FH13] Nikos Frantzikinakis and Bernard Host, Uniformity of Multiplica-tive Functions and Partition Regularity of Some Quadratic Equations ,ArXiv e-prints (2013).[Gra08] Ron Graham,
Old and New Problems in Ramsey Theory , Bolyai Soc.math. Stud. (2008), 105–118.[MP] Kellen Myers and Joseph Parrish, Some Nonlinear Rado Numbers , (inpreparation).[Rad33] Richard Rado,
Studien zur Kombinatorik , Math. Zeit. (1933), 242–280.[Sch16] Issai Schur, Uber die Kongruenz x m + y m ≡ z m ( mod p ), Jahresber.Deutsche Math.-Verein.25