How does a Bitcoin client know its transaction has been accepted by the whole network?
6 blocks, not 6 clients. The chances of catching up to the block with the transactions that is buried 6 blocks deep is slim. Granted of course that the dishonest party has not a very significant part (say less than 15% or so) of the hashing power. These changes can be calculated using Poisson distribution, as done in Satoshi's original paper (last pages).