Independent finite sums in graphs defined on the natural numbers

Tomasz Luczak, Vojtěch Rödl, Tomasz Schoen
1998 Discrete Mathematics  
In this note we present several results related to conjectures of Erd6s and Hajnal on the existence of independent sets with good arithmetic properties in a locally sparse graph whose vertices are natural numbers. In particular, we prove that if k, f >~ 2 and a graph G defined on the natural numbers contains no copies of the complete graph on k vertices, then there exists a subset A C ~ such that the set FS~ no the following holds: for each graph G with vertex set { 1,2 ..... n} which contains
more » ... o copies of the complete graph Kk on k vertices there exists A C_{1 ..... n} such that all finite sums of different elements of A span in G an independent set. An infinite version of this problem was stated by Andr~ts Hajnal, who asked if, for a graph G defined on the set of natural numbers, there exists an infinite set A with the above property. Hajnal's question has been recently answered in the negative by Deuber, Gunderson, Hindman and Strauss in [1]. In the same paper the authors prove also that disjoint and bipartite versions of Hajnal's conjecture hold (see [1] for details). The main result of this note, Theorem 5, asserts that a finite (or, more precisely, 'semi-infinite') version of Hajnal's conjecture remains true as well, which, in particular, settles Erd6s' conjecture in the affirmative. We also provide a simple proof for a bipartite version of Hajnal's question, stated as Theorem 7.
doi:10.1016/s0012-365x(97)00064-2 fatcat:qh27gy4bjjaf5gzexnuflwot3m