2 research outputs found
3-color Schur numbers
Let k ≥ 3 be an integer, the Schur number Sk(3) is the least positive integer, such that for
every 3-coloring of the integer interval [1, Sk(3)] there exists a monochromatic solution to
the equation x1+ · · · + xk= xk+1, where xi
, i = 1, . . . , k need not be distinct.
In 1966, a lower bound of Sk(3) was established by Znám (1966). In this paper, we
determine the exact formula of Sk(3) = k
3 + 2k
2 − 2, finding an upper bound which
coincides with the lower bound given by Znám (1966). This is shown in two different
ways: in the first instance, by the exhaustive development of all possible cases and in the
second instance translating the problem into a Boolean satisfiability problem, which can
be handled by a SAT solver
Rado Numbers and SAT Computations
Given a linear equation , the -color Rado number
is the smallest integer such that every -coloring of
contains a monochromatic solution to . The
degree of regularity of , denoted , is the largest
value such that is finite. In this article we present new
theoretical and computational results about the Rado numbers
and the degree of regularity of three-variable equations .
% We use SAT solvers to compute many new values of the three-color Rado
numbers for fixed integers and . We also give a
SAT-based method to compute infinite families of these numbers. In particular,
we show that the value of is equal to for
. This resolves a conjecture of Myers and implies the conjecture that
the generalized Schur numbers
equal for . Our SAT solver computations, combined with
our new combinatorial results, give improved bounds on and
exact values for . We also give counterexamples to a
conjecture of Golowich