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COMBmA'romcA 7 (2) (1987) 151--160 SMALLEST n-UNIFORM HYPERGRAPH WITH
 

Summary: THE
COMBmA'romcA 7 (2) (1987) 151--160
SMALLEST n-UNIFORM HYPERGRAPH WITH
POSITIVE DISCREPANCY
N. ALON, D. J. KLEITMAN, C. POMERANCE,
M. SAKS and P. SEYMOUR
Received 15 March 1986
A two-coloring 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 two-coloring 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 n-uniform 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 two-coloring
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

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University
Pomerance, Carl - Department of Mathematics, Dartmouth College

 

Collections: Mathematics