 
Summary: Europ . J. Combinatories (1985) 6, 201203
An Application of Graph Theory to Additive Number Theory
NOGA ALON* AND P. ERDOS
A sequence of integers A = {a, < a2 < · · . < a,,} is a Bzk) sequence if the number of representa
tions of every integer as the sum of two distinct a,s is at most k . In this note we show that every
B' sequence of n terms is a union of
C2(k) · n113 BzB (1) sequences, aand tatthe re is aB (k) sequence
of n terms which is not a union of cik · n'"3 Bz" sequences. This solves a problem raised in
2
2
[3, 4]. Our proof uses some results from extremal graph theory . We also discuss some related
problems and results .
Sidon called a finite or infinite sequence of integers A= {a, < a 2 < } a B3kl sequence
if the number of representations of every integer as the sum of two distinct a,s is at most
k. In particular he was interested in B2", or, for short, B2 sequences, i.e. the case where
all the sums ai + a; are distinct.
Let f,, denote the maximal cardinality of a B2 subsequence of {1, 2, . . . , n}. Turán and
Erds proved [5]
n t/2  O(n5/16)
