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Source: 12-th Hungary-Israel Binational Mathematical Competition 2001

April 19, 2007

graph theorycombinatorics unsolvedcombinatorics

Problem Statement

Here

G_{n}

denotes a simple undirected graph with

n

vertices,

K_{n}

denotes the complete graph with

n

vertices,

K_{n,m}

the complete bipartite graph whose components have

m

and

n

vertices, and

C_{n}

a circuit with

n

vertices. The number of edges in the graph

G_{n}

is denoted

e(G_{n})

. (a) If

G_{n}

does not contain

K_{2,3}

, prove that

e(G_{n}) \leq\frac{n\sqrt{n}}{\sqrt{2}}+n

. (b) Given

n \geq 16

distinct points

P_{1}, . . . , P_{n}

in the plane, prove that at most

n\sqrt{n}

of the segments

P_{i}P_{j}

have unit length.

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