Improving Gebauer’s Construction of 3-Chromatic Hypergraphs with Few Edges

Author Jakub Kozik

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Author Details

Jakub Kozik
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

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Jakub Kozik. Improving Gebauer’s Construction of 3-Chromatic Hypergraphs with Few Edges. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 89:1-89:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In 1964 Erdős proved, by randomized construction, that the minimum number of edges in a k-graph that is not two colorable is O(k² 2^k). To this day, it is not known whether there exist such k-graphs with smaller number of edges. Known deterministic constructions use much larger number of edges. The most recent one by Gebauer requires 2^{k+Θ(k^{2/3})} edges. Applying a derandomization technique we reduce that number to 2^{k+Θ̃(k^{1/2})}.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypergraphs
  • Mathematics of computing → Probabilistic algorithms
  • Theory of computation → Pseudorandomness and derandomization
  • Property B
  • Hypergraph Coloring
  • Deterministic Constructions


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  1. József Beck. On 3-chromatic hypergraphs. Discrete Mathematics, 24(2):127-137, 1978. URL:
  2. Danila D. Cherkashin and Jakub Kozik. A note on random greedy coloring of uniform hypergraphs. Random Structures & Algorithms, 47(3):407-413, 2015. URL:
  3. Paul Erdős. On a combinatorial problem. II. Acta Mathematica Academiae Scientiarum Hungaricae, 15:445-447, 1964. Google Scholar
  4. Ofer Gabber and Zvi Galil. Explicit constructions of linear-sized superconcentrators. J. Comput. System Sci., 22(3):407-420, 1981. Special issued dedicated to Michael Machtey. URL:
  5. Heidi Gebauer. On the construction of 3-chromatic hypergraphs with few edges. Journal of Combinatorial Theory. Series A, 120(7):1483-1490, 2013. URL:
  6. Nathan Linial, Michael Luby, Michael Saks, and David Zuckerman. Efficient construction of a small hitting set for combinatorial rectangles in high dimension. Combinatorica, 17(2):215-234, 1997. URL:
  7. G. A. Margulis. Explicit constructions of expanders. Problemy Peredači Informacii, 9(4):71-80, 1973. Google Scholar
  8. Jaikumar Radhakrishnan and Aravind Srinivasan. Improved bounds and algorithms for hypergraph 2-coloring. Random Structures & Algorithms, 16(1):4-32, 2000. URL:<4::AID-RSA2>3.3.CO;2-U.
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