Problem 4

There is an integer

n > 1

. There are

n^2

stations on a slope of a mountain, all at different altitudes. Each of two cable car companies,

A

and

B

, operates

k

cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The

k

cable cars of

A

have

k

different starting points and

k

different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for

B

. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer

k

for which one can guarantee that there are two stations that are linked by both companies.

Step 4 of 8: Find a long chain

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In plain words

If every A-chain had at most n stations, n−1 chains could hold at most n²−n stations, fewer than all n² stations.

|C|\ge n+1\quad\text{for some A-chain }C.

Detailed analysis

Suppose every A-chain had at most n vertices. Since there are n−1 A-chains, they would contain at most n(n−1)=n²−n stations, contradicting the existence of n² stations. Hence some A-chain C contains at least n+1 stations.