On the density of 2-colorable 3-graphs in which any four points span at most two edges
✍ Scribed by Klas Markström; John Talbot
- Publisher
- John Wiley and Sons
- Year
- 2009
- Tongue
- English
- Weight
- 161 KB
- Volume
- 18
- Category
- Article
- ISSN
- 1063-8539
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
Let ex~2~(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)$\end{document}. We improve the previously best known lower and upper bounds of 0.25682 and 3/10−ε, respectively, by showing that
This implies the following new upper bound for the Turán density of K
In order to establish these results we use a combination of the properties of computer‐generated extremal 3‐graphs for small n and an argument based on “super‐saturation”. Our computer results determine the exact values of ex(n, K) for n≤19 and ex~2~(n, K) for n≤17, as well as the sets of extremal 3‐graphs for those n. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 105–114, 2010