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(1964)
On a graph problem of Turán (In Hungarian)
- Publisher
- Springer Journals
- Copyright
- Copyright © Science China Press 2024
- ISSN
- 1674-7283
- eISSN
- 1869-1862
- DOI
- 10.1007/s11425-023-2196-x
- Publisher site
- See Article on Publisher Site
Abstract
Let r ⩾ 2 be an integer. The real number α ∈ [0, 1) is a jump for r if there exists a constant c > 0 such that for any ϵ > 0 and any integer m ⩾ r, there exists an integer n0(ϵ, m) satisfying any r-uniform graph with n ⩾ n0 (ϵ, m) vertices and density at least α + ϵ contains a subgraph with m vertices and density at least α + c. A result of Erdős and Simonovits (1966) and Erdős and Stone (1946) implies that every α ∈ [0, 1) is a jump for r = 2. Erdős (1964) asked whether the same is true for r ⩾ 3. Frankl and Rödl (1984) gave a negative answer by showing that 1−1ℓr−1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1-{1 \over {\ell^{r-1}}}$$\end{document} is not a jump for r if r ⩾ 3 and ℓ > 2r. After that, more non-jumps are found by using a method of Frankl and Rödl (1984). Motivated by an idea of Liu and Pikhurko (2023), in this paper, we show a method to construct maps f: [0, 1) → [0, 1) that preserve non-jumps, i.e., if α is a non-jump for r given by the method of Frankl and Rödl (1984), then f(α) is also a non-jump for r. We use these maps to study hypergraph Turán densities and answer a question posed by Grosu (2016).
Journal
Science China Mathematics – Springer Journals
Published: Dec 1, 2024
Keywords: jumping number; non-jump; Turán density; hypergraph Lagrangian; 05C35; 05C65; 05D05