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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)
https://doi.org/10.4230/LIPIcs.ICALP.2021.89

Abstract

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
Keywords
  • Property B
  • Hypergraph Coloring
  • Deterministic Constructions

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References

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