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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-3/17g/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 17G / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

The diagonal Ramsey number

R (k)

is the least

n

such that every red-blue coloring of

K_{n}

contains a monochromatic

K_{k}

. The standard recursion

R (s, t) \leq R (s - 1, t) + R (s, t - 1)

gives

R (k) \leq (k - 1 2 k - 2) < 4^{k} .

Thus the converted

k^{4}

is a superscript-order error; that bound is false for large

k

.

In a red-blue coloring, if the red spanning graph is connected it has a red spanning tree. Otherwise its red components are joined pairwise by blue edges, making the blue graph connected, so it has a blue spanning tree.

The result fails for three colors. Color the six edges of

K_{4}

by its three perfect matchings, one color per matching. Every monochromatic graph then consists of two disjoint edges and has no spanning tree.

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