Összegmentes halmaz

Az additív kombinatorika és számelmélet területén egy G Abel-csoport A részhalmaza akkor összegmentes (sum-free), ha az A⊕A összeghalmaz és az A diszjunktak. Más megfogalmazásban, A akkor összegmentes, ha az $a+b=c$ egyenletnek nincs megoldása $a,b,c\in A$ -ban.

Például a páratlan számok az egész számok összegmentes részhalmaza, vagy az {1,...,N} (N páros) számok nagy összegmentes részhalmaza az {N/2+1, ..., N} halmaz. A Fermat–Wiles-tétel úgy is megfogalmazható, hogy a nemnulla n-edik hatványok az n > 2 egészek összegmentes részhalmazát alkotják.

Néhány alapvető, az összegmentes halmazokkal kapcsolatos kérdés:

Egy N egészre az {1, ..., N} halmaznak hány összegmentes részhalmaza létezik? Ben Green megmutatta,^[1] hogy a válasz $O(2^{N/2})$ , ahogy azt a Cameron–Erdős-sejtés^[2] megjósolta (lásd Sloane: A007865).
Egy G Abel-csoportnak hány összegmentes részhalmaza létezik?^[3]
Mekkora a legnagyobb összegmentes részhalmaza egy G Abel-csoportnak?^[3]

Egy összegmentes halmaz akkor maximális, ha nem valódi részhalmaza egy másik összegmentes halmaznak.

Jegyzetek

↑ Ben Green, The Cameron–Erdős conjecture, Bulletin of the London Mathematical Society 36 (2004) pp.769-778
↑ P.J. Cameron and P. Erdős, On the number of sets of integers with various properties, Number theory (Banff, 1988), de Gruyter, Berlin 1990, pp.61-79
↑ ^a ^b Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.

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