Coloring H-free Hypergraphs
Tom Bohman, Alan Frieze, Dhruv Mubayi

TL;DR
This paper investigates the minimum edges needed in hypergraphs avoiding certain substructures while having high chromatic number, providing new bounds and generalizations for various hypergraph classes.
Contribution
It introduces improved bounds for the minimum edges in hypergraphs with forbidden configurations and high chromatic number, extending classical results to hypergraphs.
Findings
Bound for $(r,l)$-systems with high chromatic number improves previous results.
Minimum edges in hypergraphs with independent neighborhoods grow as ilde{k}^{r+1/(r-1)}.
Chromatic number of $T$-free hypergraphs is polynomial in the size of the hypertree.
Abstract
Fix and a collection of -uniform hypergraphs . What is the minimum number of edges in an -free -uniform hypergraph with chromatic number greater than . We investigate this question for various . Our results include the following: An -system is an -uniform hypergraph with every two edges sharing at most vertices. For sufficiently large, the minimum number of edges in an -system with chromatic number greater than is at most , where This improves on the previous best bounds of Kostochka-Mubayi-R\"odl-Tetali \cite{KMRT}. The upper bound is sharp aside from the constant as shown in \cite{KMRT}. The minimum number of edges in an -uniform hypergraph with independent neighborhoods and chromatic number greater than is of order as . This…
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Taxonomy
TopicsGraph Labeling and Dimension Problems · graph theory and CDMA systems · Advanced Graph Theory Research
