2015/05/19 by Tristin Lehmann, Lehmann, Tristin, Donald L. Vestal +2
Mathematics · #05D10 #Combinatorics (math.CO) #FOS: Mathematics #advanced mathematical theories #math.CO #msc:05D10
paper · pdf · doi:10.48550/arxiv.1505.05071
Keywords: Ramsey Theory, Rado Number
arxiv created 2015/05/19 · openalex publication_date 2015/05/19 · arxiv updated 2015/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a positive integer m and a real number c, let R = R(m,c,2) denote the discrete 2-color Rado number for the equation x1 + x2 + … + xm + c = 2x0. In other words, R is the smallest integer such that for any coloring of the integers 1, 2, …, R, there exist numbers x1, x2, …, xm, x0, all with the same color, such that x1 + x2 + … + xm + c = 2x0. In this article we show that if m ≥ 2 and c > 0, then R(m,c,2) = \begincases ∞ · for m even, c odd \newline \lceil (m)/(2) \lceil (m+c)/(2) \rceil + (c)/(2) \rceil · otherwise. \endcases For real numbers a and c, we look at the 2-color Rado number for the equation x1 + c = ax0. We show that if a > 1 and c > 0, then the 2-color continuous Rado number is R_ℝ(1, c, a) = \begincases ∞ if a=1 \newline (c)/(a-1) otherwise. \endcases From this, we will show that the discrete Rado number is R(1, c, a) = \begincases (c)/(a-1) if ( a-1 ) | c \newline ∞ otherwise. \endcases