A sum packing problem of Erd\H{o}s

and the Conway-Guy sequence

Author:
Tom Bohman

Journal:
Proc. Amer. Math. Soc. **124** (1996), 3627-3636

MSC (1991):
Primary 11P99; Secondary 05D10

DOI:
https://doi.org/10.1090/S0002-9939-96-03653-2

MathSciNet review:
1363448

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Abstract | References | Similar Articles | Additional Information

Abstract: A set of positive integers has distinct subset sums if the set has distinct elements. Let

In 1931 Paul Erd\H{o}s conjectured that for some constant . In 1967 John Conway and Richard Guy constructed an interesting sequence of sets of integers. They conjectured that these sets have distinct subset sums and that they are close to the best possible (with respect to largest element). We prove that sets from this sequence have distinct subset sums. We also present some variations of this construction that give microscopic improvements in the best known upper bound on .

**[AS]**Peter M. Lee,*Bayesian statistics*, A Charles Griffin Book, Edward Arnold, London; copublished in the Americas by Halsted Press [John Wiley & Sons, Inc.], New York, 1992. An introduction; Reprint of the 1989 original. MR**1182312****[B]**T. Bohman, A Construction For Sets of Integers With Distinct Subset Sums, in preparation.**[CG]**J.H. Conway and R.K. Guy, Sets of natural numbers with distinct sums,*Notices Amer. Math. Soc.***15**(1968), 345.**[E1]**P. Erd\H{o}s, personal communication.**[E2]**P. Erd\H{o}s, Problems and results from additive number theory,*Colloq. Théorie des Nombres, Bruxelles*, 1955, Liège & Paris, 1956, 127-137, esp. p. 137. MR**18:18a****[G1]**Richard K. Guy,*Sets of integers whose subsets have distinct sums*, Theory and practice of combinatorics, North-Holland Math. Stud., vol. 60, North-Holland, Amsterdam, 1982, pp. 141–154. MR**806978****[G2]**R. K. Guy,*Unsolved Problems in Number Theory*, 2nd ed., Springer-Verlag, New York, 1994. CMP**95:02****[L]**W. F. Lunnon,*Integer sets with distinct subset-sums*, Math. Comp.**50**(1988), no. 181, 297–320. MR**917837**, https://doi.org/10.1090/S0025-5718-1988-0917837-5

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Additional Information

**Tom Bohman**

Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Address at time of publication:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
bohman@math.rutgers.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03653-2

Received by editor(s):
June 6, 1995

Communicated by:
Jeffry N. Kahn

Article copyright:
© Copyright 1996
American Mathematical Society