By S. Gill Williamson

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**Example text**

L,where L is an integer 2: 2,which has the following two properties: 1) All intervals Av have the same length bv - a// == w, and their open kernels are pairwise disjoint. 2) The initial elements a v , // == 1, ... , L, form an arithmetic progression with al < .... < aL. Because of 1) the same then also holds for the final elements bv , v == 1, ... ,n. Again we call the step-length of the arithmetic progression {aI, ... , aL} also the step-length of the arithmetic interval sequence. - the shrink factor of {AI, ...

Althofer et al. ), Numbers, Information and Complexity, 45-74. © 2000 Kluwer Academic Publishers. 46 of subsets of a finite set. The ingenious, relatively elementary methods were elaborated by Ahlswede and Khachatrian in several papers [2, 1, 4, 3]. Our aim is to provide a unifying approach such that most of the results are covered. Since Erdos, Ko and Rado [11] have initiated the study of such problems in the thirties many results were obtained by several authors. Here we cite only the recent papers which are related to the new AK-methods.

A 76, 1996, 99-107. [4) E. Harzheim, "Weakly arithmetic progressions in sets of natural numbers", Discrete Math. 89, 1991, 105-107. [5) E. Harzheim, "On weakly arithmetic progressions", Discrete Math. 138, 1995, 255-260. B. Nathanson, "Arithmetic progressions contained in sequences with bounded gaps", Ganad. Math. Bull 23, 1980, 491-493. R. Rabung, "On applications of van der Waerden's theorem", Math. Mag. 48, 1975, 142-148. [8) A. Sarkozy, "Some metric problems in the additive number theory I", Annales Univ.