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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 209.28003

**Autor: ** Erdös, Paul

**Title: ** On a lemma of Hajnal-Folkman (In English)

**Source: ** Combinat. Theory Appl., Colloquia Math. Soc. János Bolyai 4, 311-316 (1970).

**Review: ** [For the entire collection see Zbl 205.00201.]

The symbol (m,n,i,r) ––> p means that if |*S*| = m \geq n, A_{j} \subset *S*, |A_{j}| \geq n is any family of subsets of *S* which cannot be represented by any i elements of *S*, then there is a subset *S*_{1} of *S*, |*S*_{1}| = p, p \geq r, every r-tuple of which occurs in some A_{j}. (m,n,i,r) (not)––> p means that (m,n,i,r) ––> p does not hold. (2n-1,n,1,2) ––> n+1 is an old result of Hajnal and Folkman. The author proves (2n+i-2,n,i,2) ––> n+i and that this result is best possible. Several problems are posed, some of which have been settled since. Also a connection with Ramsey's theorem is established.

**Classif.: ** * 05C55 Generalized Ramsey theory

**Citations: ** Zbl 205.002(EA)

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