TY - JOUR
TI - Computational advances in Rado numbers
DO - https://doi.org/doi:10.7282/T3GH9KTT
PY - 2015
AB - In this dissertation, we present new methods in the computation of Rado numbers. These methods are applied to several families of equations. The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. For any particular equation E, the Rado number R_r(E) is the smallest N such that any r-coloring chi:{1,2,...,N} -> {1,2,...,r} must induce a monochromatic solution to E. We will lay out the history of this field and provide some structure as context for new results. Then we will discuss the new methods and computational tools that provide the foundation of the thesis. The 2-color Rado numbers R_2(2x+2y+kz = 3w) and R_2(kx+(k+1)y = (k+2)z) are computed for small values of the parameter k. The 2-color off-diagonal Rado numbers R_2(x + ay = z; x + by = z) are provided for 1 <= a, b <= 20. Likewise, 3-color off-diagonal Rado numbers R_2(x + y = az; x + y = bz; x + y = cz) are computed for 1 <= a, b, c <= 5. We confirm the long-standing conjecture that the 3-color generalized Schur numbers R_3(x_1 + x_2 + ... + x_{m-1} = xm ) are m^3 - m^2 - m - 1 for m = 7, 8, 9, 10 (effectively doubling the empirical evidence for the conjecture) and provide the related Rado numbers R_3( x1 + ... + x(m-2) + k x(m-1) = xm ) for certain (k,m) values. We prove a lower bound for the r-color non-homogeneous Schur numbers: R_r( x + y + c = z) >= (3^r - 1)(c+1)/2 for c >= 0. We also compute the precise values for r = 4 and -20 <= c <= 7 and generalize this bound for m >= 3 variables. We provide the 2-color Rado numbers for 1/x + 1/y = 1/z and a few other equations involving reciprocals. We also construct a coloring proving R_2(x^2 + y^2 = z^2) > 6500. (It is not known whether this Rado number is finite.) We compute the 2- and 3-color Rado numbers for other sums-of-squares equations, sum_{i=1}^a x_i^2) = sum_{i=1}^b y_i^2, and we prove a universal upper bound for a <= b <= ca for a constant c between 1 and 2 (different values of c give different upper bounds). We follow this with Rado numbers for other assorted families of quadratic equations. We also present quantitative analogues of Hindman's theorem, which guarantees monochromatic solutions to systems like {x+y+z = w; x*y*z = v}. We conclude by suggesting a number of conjectures, extensions, and generalizations of these results for future work.
KW - Mathematics
KW - Ramsey theory
LA - eng
ER -