Geodesic
Remarks on a Problem of Moser
Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 19-21

Voir la notice de l'article provenant de la source Cambridge University Press

Let M(n) be the set of all the points (x 1, x 2,..., x n)∈E n such that x i∈{0,1,2} for each i= 1, 2,..., n and let f(n) be the cardinality of a largest subset of M(n) containing no three distinct collinear points. L. moser [4] asked for a proof of the inequality Let us consider the set Sn of those points (x 1x 2,..., x n)∈M(n) which satisfy |{i:X i= 1}| = [(n +1)/3]. As S n is a subset of the sphere with center at (1, 1,..., 1) and radius (n-[(n+1)/3])1/2, no three distinct points of S n are collinear. Thus we have 1 
Chvátal, V. Remarks on a Problem of Moser. Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 19-21. doi: 10.4153/CMB-1972-004-8
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[1] 1. V. Chvátal, Some unknown Van der Waerden numbers, Combinatorial structures and their applications (Guy, R. K. et al., ed.), Gordon and Breach, New York (1970), 31-33. Google Scholar

[2] 2. Hales, A. W. and Jewett, R. I., Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222-229. Google Scholar

[3] 3. Moser, L., Problem 21, Proc. of Number Theory Conference, Univ. of Colorado, 1963, Mimeographed, 79. Google Scholar

[4] 4. Moser, L., Problem P. 170, Canad. Math. Bull. 13 (1970), p. 268. Google Scholar

[5] 5. Roth, K. F., On certain sets of integers, J. Londo. Math. Soc. 28 (1953), 104-109. Google Scholar

[6] 6. Szemerédi, E., On sets of integers containing no four elements in arithmetic progression, Acta. Math. Acad. Sci. Hungar. 20 (1969), 89-104. Google Scholar

[7] 7. Van der Waerden, B. L., Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1928), 212-216. Google Scholar

[1] 1. V. Chvátal, Some unknown Van der Waerden numbers, Combinatorial structures and their applications (Guy, R. K. et al., ed.), Gordon and Breach, New York (1970), 31-33. Google Scholar

[2] 2. Hales, A. W. and Jewett, R. I., Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222-229. Google Scholar

[3] 3. Moser, L., Problem 21, Proc. of Number Theory Conference, Univ. of Colorado, 1963, Mimeographed, 79. Google Scholar

[4] 4. Moser, L., Problem P. 170, Canad. Math. Bull. 13 (1970), p. 268. Google Scholar

[5] 5. Roth, K. F., On certain sets of integers, J. Londo. Math. Soc. 28 (1953), 104-109. Google Scholar

[6] 6. Szemerédi, E., On sets of integers containing no four elements in arithmetic progression, Acta. Math. Acad. Sci. Hungar. 20 (1969), 89-104. Google Scholar

[7] 7. Van der Waerden, B. L., Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1928), 212-216. Google Scholar

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