Rado's theorem (Ramsey theory)
- Rado's theorem (Ramsey theory)
: "There is also a Rado's theorem about harmonic functions."
Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the English mathematician Richard Rado.
Let "Ax=0" be a system of linear equations, where "A" is a matrix with integer entries. This system is said to be "r-regular" if, for every "r"-coloring of the natural numbers 1, 2, 3, ..., the system has a monochromatic solution. A system is "regular" if it is "r-regular" for all
Rado's theorem states that a system "Ax=0" is regular if and only if the matrix "A" satisfies the "columns condition". Let "ci" denote the "i"-th column of "A". The matrix "A" satisfies the columns condition provided that there exists a partition of the column indices "C1", "C2", ..., "Cn" such if , then
# "s1 = 0"
# for all , "si" can be written as a linear combination of the "cj"'s in the "Ck" with "k"<"i".
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