Geodesic
Monochromatic Homothetic Copies of {1, 1 + s, 1 + s + t}
Canadian mathematical bulletin, Tome 40 (1997) no. 2, pp. 149-157

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

For positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1, 2,...,N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}.We show that f (s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden’s theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + t}. We also show that f (s, t) = 4(s + t) + 1 in many other cases (for example, whenever s > 2t > 2 and t does not divide s), and that f (s, t) ≤ 4 (s + t) + 1 for all s, t.Thus the set of homothetic copies of {1, 1 + s, 1 + s + t} is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other “natural” sets of triples, quadruples, etc., have simple (or easily estimated) Ramsey functions.
DOI : 10.4153/CMB-1997-018-3
Mots-clés : 05D10 
Brown, Tom C.; Landman, Bruce M.; Mishna, Marni. Monochromatic Homothetic Copies of {1, 1 + s, 1 + s + t}. Canadian mathematical bulletin, Tome 40 (1997) no. 2, pp. 149-157. doi: 10.4153/CMB-1997-018-3
@article{10_4153_CMB_1997_018_3,
     author = {Brown, Tom C. and Landman, Bruce M. and Mishna, Marni},
     title = {Monochromatic {Homothetic} {Copies} of {1, 1 + s, 1 + s + t}},
     journal = {Canadian mathematical bulletin},
     pages = {149--157},
     year = {1997},
     volume = {40},
     number = {2},
     doi = {10.4153/CMB-1997-018-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1997-018-3/}
}
TY  - JOUR
AU  - Brown, Tom C.
AU  - Landman, Bruce M.
AU  - Mishna, Marni
TI  - Monochromatic Homothetic Copies of {1, 1 + s, 1 + s + t}
JO  - Canadian mathematical bulletin
PY  - 1997
SP  - 149
EP  - 157
VL  - 40
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1997-018-3/
DO  - 10.4153/CMB-1997-018-3
ID  - 10_4153_CMB_1997_018_3
ER  - 
%0 Journal Article
%A Brown, Tom C.
%A Landman, Bruce M.
%A Mishna, Marni
%T Monochromatic Homothetic Copies of {1, 1 + s, 1 + s + t}
%J Canadian mathematical bulletin
%D 1997
%P 149-157
%V 40
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1997-018-3/
%R 10.4153/CMB-1997-018-3
%F 10_4153_CMB_1997_018_3

[1] 1. Brown, T. C. and Erdʺos, P. and Freedman, A. R., Quasi-progressions and descending waves, J. Combin. Theory Ser. A 53 (1990), 81–95. Google Scholar

[2] 2. Graham, R. L., Rothschild, B. L. and Spencer, J. H., Ramsey Theory, 2nd ed., John Wiley and Sons, New York, 1990. Google Scholar

[3] 3. Landman, Bruce M. and Greenwell, Raymond N., Values and bounds for Ramsey numbers associated with polynomial iteration, Discrete Math. 68 (1988), 77–83. Google Scholar

[4] 4. Landman, Bruce M. and Greenwell, Raymond N., Some new bounds and values for van der Waerden-like numbers, Graphs Combin. 6 (1990), 287–291. Google Scholar

[5] 5. van der Waerden, B. L., Beweis einer Baudetschen Vermutung, Nieuw Arch.Wisk. 15 (1927), 212–216. Google Scholar

Cité par Sources :