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On sets of integers not containing arithmetic progressions of prescribed length

Published online by Cambridge University Press:  09 April 2009

H. L. Abbott ,
A. C. Liu  and
J. Riddell
H. L. Abbott
Affiliation:
University of Alberta, Edmonton, Alberta
A. C. Liu
Affiliation:
University of Alberta, Edmonton, Alberta
J. Riddell
Affiliation:
University of Victoria, Victoria, British Columbia
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Let m, n and l be positive integers satisfying m ≦ n ≦ l ≦ 3. Denote by h(m, n, l) the largest integer with the property that from every n-subset of {1,2, …, m} one can select h(m, n, l) integers no l of which are in arithmetic progression. Let f(n, l) = h(n, n, l) and let g(n, l) = minmh(m, n, l). In what follows, by a P1-free set we shall mean a set of integers not containing an arithmetic progression of length l.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 2 , September 1974 , pp. 188 - 193
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Abbott, H. L. and Liu, A. C., ‘On partitioning integers into progression free sets’, J. Comb. T. (to appear).Google Scholar
[2]Behrena, F. A., ‘On sets of integers which contain no three terms in arithmetical progression’, Proc. Nat. Acad. Sci. U. S. A. 32 (1946), 331–332.CrossRefGoogle Scholar
[3]Erdös, P., ‘Some recent advances and current problems in number theory’, Lectures on Modern Mathematics, Vol. 3, Saaty, T. L., ed. (Wiley, New York, 1963).Google Scholar
[4]Lorentz, G. G., ‘On a problem of additive number theory’, Proc. Amer. Math. Soc. 5 (1954), 838–841.CrossRefGoogle Scholar
[5]Moser, L., ‘On non-averaging sets of integers’, Can. J. Math. 5 (1953), 245–252.CrossRefGoogle Scholar
[6]Rankin, R. A., ‘Sets of integers containing not more than a given number of terms in arithmetical progression’, Proc. Roy. Soc. Edin, A, 65 (1962) 332–344.Google Scholar
[7]Riddell, J., ‘On sets of integers containing no l terms in Arithmetic progression’, Neiuw Arch. voor Wisk. (3), 17 (1969), 204–209.Google Scholar
[8]Roth, K. F., ‘Sur quelques ensembles d'entiers’, C. R. Acad. Sci. Paris. 234 (1952), 388–390.Google Scholar
[9]Salem, R. and Spencer, D. C., ‘On sets of integers which contain no three terms in arithmetical progression’, Proc. Nat. Acad. Sci. U. S.A. 28 (1942), 561–563.CrossRefGoogle ScholarPubMed
[10]Szemerédi, E., ‘On sets of integers containing no four terms in arithmetic progression’, Acta. Math. Acad. Sci. Hung. 20 (1969), 89–104.CrossRefGoogle Scholar
[11]Wagstaff, S. S. Jr, ‘On sequences of integers with no 4, or no 5 numbers in arithmetical progression’, Math. Comp. 21 (1967), 695–699.CrossRefGoogle Scholar