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**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 a_{k}, 1 \leq k \leq m define an m-cube Q_{m} to be the set **{**a+**sum**^{m}_{k = 1}\epsilon_{k}a_{k}: \epsilon_{k} = 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 C_{k}, 1 \leq k \leq r, some C_{i} 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,a_{1},a_{2})-**{**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 **{****sum**^{oo}_{k = 1}\epsilon_{k}a_{k}: \epsilon_{k} = 0 or 1 with 0 \leq **sum**^{oo}_{k = 1}\epsilon_{k} < 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))r^{2}. 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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag