Ramsey partitions of integers and fair divisions (Q1196689)

Let \(k_ 1,\) \(k_ 2\) be positive integers, the partition \({\mathcal P}=(\alpha_ 1,\alpha_ 2,\dots,\alpha_ n)\) of \(k_ 1+k_ 2\) is called Ramsey partition for \(k_ 1\), \(k_ 2\) if for any sublist \({\mathcal L}\) of \({\mathcal P}\) either there is a sublist of \({\mathcal L}\) which sums to \(k_ 1\), or a sublist of \({\mathcal P}-{\mathcal L}\) which sums to \(k_ 2\). It is shown that there is a unique Ramsey partition for \(k_ 1\), \(k_ 2\) having the smallest number \(n\) of terms and that \(n\) is one more the sum of quotients in the Euclidean algorithm for \(k_ 1\), \(k_ 2\). An application to the fair division problem is also discussed. Suppose two persons want to divide a cake fairly in the ratio \(k_ 1\): \(k_ 2\). It can be done trivially using \(k_ 1+k_ 2-1\) cuts. Every Ramsey partition of \(k_ 1+k_ 2\) yields a fair division with fewer cuts except for \(k_ 1=1\), \(k_ 2=1,2,4\).