The smallestn-uniform hypergraph with positive discrepancy


A two-coloring of the verticesX of the hypergraphH=(X, ε) by red and blue hasdiscrepancy d ifd is the largest difference between the number of red and blue points in any edge. A two-coloring is an equipartition ofH 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 ann-uniform hypergraph (all edges have sizen) having positive discrepancy. Erdős and Sós asked: isf(n) unbounded? We answer this question in the affirmative and show that there exist constantsc 1 andc 2 such that

$$\frac{{c_1 \log (snd(n/2))}}{{\log \log (snd(n/2))}} \leqq f(n) \leqq c_2 \frac{{\log ^3 (snd(n/2))}}{{\log \log (snd(n/2))}}$$

where snd(x) is the least positive integer that does not dividex.

This is a preview of subscription content, access via your institution.


  1. [1]

    N. Alon andK. Berman, Regular hypergraphs, Gordon’s lemma, steinitz’ lemma and invariant theory,to appear in J. Combinatorial Theory, Ser. A.

  2. [2]

    J. Beck, Balanced 2-colorings of finite sets in the square I,Combinatorica,1 (1981), 327–335.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    J. Beck andT. Fiala, “Integer-making” theorems,Discrete Applied Math,3 (1981), 1–8.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    P. Erdős andV. T. Sós,private communication.

  5. [5]

    J. E. Graver, A survey of the maximum depth problem for indecomposable exact covers,in: Infinite and Finite Sets, Col. Math. János Bolyai, (1973) 731–743.

  6. [6]

    A. E. Ingham,The Distribution of Prime Numbers, Cambridge Tracts in Mathematics, no. 30 Cambridge University Press, 1937.

  7. [7]

    L. Lovász, J. Spencer andK. Vesztergombi, Discrepancy of set-systems and matrices,preprint.

  8. [8]

    W. Rudin,Principles of Mathematical Analysis. McGraw Hill, 1964.

  9. [9]

    V. T. Sós, Irregularities of partitions: Ramsey theory, uniform distribution,in Surveys in Combinatorics, London Math. Soc. Lecture Notes Series82 (1983), 201–246.

  10. [10]

    J. Sencer, Six standard deviations suffice,Trans. Amer. Math. Soc.,289 (1985), 679–706.

    Article  MathSciNet  Google Scholar 

Download references

Author information



Additional information

The first two authors were supported in part by NSF under grant DMS 8406100; the third one was supported in part by NSF under grant DCR 8421341.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alon, N., Kleitman, D.J., Pomerance, C. et al. The smallestn-uniform hypergraph with positive discrepancy. Combinatorica 7, 151–160 (1987).

Download citation


  • Positive Discrepancy
  • Blue Point
  • Integer Vector
  • Prime Number Theorem
  • Number Theoretic Result