Ramsey partitions of integers and pair divisions |
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Authors: | Kevin McAvaney Jack Robertson William Webb |
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Institution: | (1) Division of Computing and Mathematics, Deakin University, 3217 Geelong, Vic., Australia;(2) Department of Mathematics, Washington State University, 99164-3113 Pullman, WA |
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Abstract: | Ifk
1 andk
2 are positive integers, the partitionP = ( 1, 2,...,
n
) ofk
1+k
2 is said to be a Ramsey partition for the pairk
1,k
2 if for any sublistL ofP, either there is a sublist ofL which sums tok
1 or a sublist ofP –L which sums tok
2. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork
1,k
2 having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork
1 andk
2.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 ratiok
1 k
2. This can be done trivially usingk
1+k
2-1 cuts. However, every Ramsey partition ofk
1+k
2 also yields a fair division algorithm. This method yields fewer cuts except whenk
1=1 andk
2=1, 2 or 4. |
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Keywords: | 05 A 17 05 D 10 11 P 81 |
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