The smallestn-uniform hypergraph with positive discrepancy

Abstract

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.

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References

  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 

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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.

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Alon, N., Kleitman, D.J., Pomerance, C. et al. The smallestn-uniform hypergraph with positive discrepancy. Combinatorica 7, 151–160 (1987). https://doi.org/10.1007/BF02579446

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Keywords

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