{"title":"Improved Lower Bound Towards Chen–Chvátal Conjecture","authors":"Congkai Huang","doi":"10.1007/s00493-025-00137-3","DOIUrl":null,"url":null,"abstract":"<p>We prove that in every metric space where no line contains all the points, there are at least <span>\\(\\Omega (n^{2/3})\\)</span> lines. This improves the previous <span>\\(\\Omega (\\sqrt{n})\\)</span> lower bound on the number of lines in general metric space, and also improves the previous <span>\\(\\Omega (n^{4/7})\\)</span> lower bound on the number of lines in metric spaces generated by connected graphs.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"86 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-025-00137-3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that in every metric space where no line contains all the points, there are at least \(\Omega (n^{2/3})\) lines. This improves the previous \(\Omega (\sqrt{n})\) lower bound on the number of lines in general metric space, and also improves the previous \(\Omega (n^{4/7})\) lower bound on the number of lines in metric spaces generated by connected graphs.
期刊介绍：
COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are
- Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups).
- Combinatorial optimization.
- Combinatorial aspects of geometry and number theory.
- Algorithms in combinatorics and related fields.
- Computational complexity theory.
- Randomization and explicit construction in combinatorics and algorithms.
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