## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  577.05007
Autor:  Brown, T.C.; Erdös, Paul; Chung, F.R.K.; Graham, Ronald L.
Title:  Quantitative forms of a theorem of Hilbert. (In English)
Source:  J. Comb. Theory, Ser. A 38, 210-216 (1985).
Review:  For positive integers m, a and ak, 1 \leq k \leq m define an m-cube Qm to be the set {a+summk = 1\epsilonkak: \epsilonk = 0 or 1, 1 \leq k \leq m}. Hilbert proved that for any positive integers m and r there exists a least integer h(m,r) such that if the set { 1,2,...,h(m,r)} is arbitrarily partitioned into r classes Ck, 1 \leq k \leq r, some Ci must contain an m cube. Schur proved that for any r, there is an s(r) so that in any partition of { 1,2,...,s(r)} into r classes some class contains a projective 2- cube Q^*2(a,a1,a2)-{0} with a = 0. This was extended by Rado for projective m-cubes and further extended by Hindman to infinite projective cubes i.e. for {sumook = 1\epsilonkak: \epsilonk = 0 or 1 with 0 \leq sumook = 1\epsilonk < oo}.
In this article the authors have investigated the function h(m,r) and several related ones. For the first interesting case m = 2 it is proved that H(2,r) = (1+0(1))r2. This result is closely related to Ramsey numbers for 4-cycles. Bounds are also obtained for deleted 2-cubes.
Reviewer:  M.Cheema
Classif.:  * 05A17 Partitions of integres (combinatorics)
05C55 Generalized Ramsey theory
Keywords:  deleted m-cube; m-cube; projective m-cubes; Ramsey numbers for 4-cycles

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