Ramsey partitions of integers and fair divisions

Ifk1 andk2 are positive integers, the partitionP = (α1,α2,...,αn) ofk1+k2 is said to be a Ramsey partition for the pairk1,k2 if for any sublistL ofP, either there is a sublist ofL which sums tok1 or a sublist ofP −L which sums tok2. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork1,k2 having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork1 andk2.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok1∶k2. This can be done trivially usingk1+k2-1 cuts. However, every Ramsey partition ofk1+k2 also yields a fair division algorithm. This method yields fewer cuts except whenk1=1 andk2=1, 2 or 4.