 
Summary: THE
COMBmA'romcA 7 (2) (1987) 151160
SMALLEST nUNIFORM HYPERGRAPH WITH
POSITIVE DISCREPANCY
N. ALON, D. J. KLEITMAN, C. POMERANCE,
M. SAKS and P. SEYMOUR
Received 15 March 1986
A twocoloring of the vertices X of the hypergraph H=(X,o°) by red and blue has dis
crepancy d if d is the largest difference between the number of red and blue points in any edge.
A twocoloring is an equipartition of H if it has discrepancy 0, i.e., every edge is exactly half red and
half blue. Letf(n) be the fewest number of edges in an nuniform hypergraph (all edges have size n)
having positive discrepancy. Erd6s and S6s asked: is f(n) unbounded? We answer this question in
the affirmative and show that there exist constants cl and c2 such that
ea log (snd (n/2)) ~ log3(snd (n/2))
log log (snd (n/2))  f (n) ~ cz log log (snd (n/Z))
where snd (x) is the least positive integer that does not divide x.
1. Introduction and Main Results
A number of recent papers have been concerned with the problem of twocoloring
the vertices of a hypergraph H=(X, 8) by red and blue so that for every edge E
of H the number of red points in E is roughly equal to the number of blue points. The
