What is the largest number of towns that can meet the following criteria?

Each pair is directly linked by just one of air, bus or train.

At least one pair is linked by air, at least one pair by bus and at least one pair by train.

No town has an air link, a bus link and a train link.

No three towns, A, B, C are such that the links between AB, AC and BC are all air, all bus or all train.

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