比赛中的闪光和彩虹

IF 1 2区 数学 Q1 MATHEMATICS Combinatorica Pub Date : 2024-04-04 DOI:10.1007/s00493-024-00090-7
António Girão, Freddie Illingworth, Lukas Michel, Michael Savery, Alex Scott
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引用次数: 0

摘要

用任意数量的颜色给顶点集 \({\{1, 2, \dotsc, n\}\}) 的完整图的边着色。如果 \(n > f(l,k)\) 那么一定存在长度为 l 的单调单色路径或长度为 k 的单调彩虹路径,那么 f(l, k) 的最小整数是多少?Lefmann、Rödl 和 Thomas 在 1992 年猜想 \(f(l, k) = l^{k - 1}\ 并证明了 \(l \ge (3 k)^{2 k}\ 的这一猜想。)我们证明了 \(l\ge k^3 (\log k)^{1 + o(1)}\) 的猜想,并建立了一般上界 \(f(l, k) \le k (\log k)^{1 + o(1)}\cdot l^{k - 1}\).这将最佳下界和上界之间的差距从指数级缩小到 k 的多项式级。
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Flashes and Rainbows in Tournaments

Colour the edges of the complete graph with vertex set \({\{1, 2, \dotsc , n\}}\) with an arbitrary number of colours. What is the smallest integer f(lk) such that if \(n > f(l,k)\) then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that \(f(l, k) = l^{k - 1}\) and proved this for \(l \ge (3 k)^{2 k}\). We prove the conjecture for \(l \ge k^3 (\log k)^{1 + o(1)}\) and establish the general upper bound \(f(l, k) \le k (\log k)^{1 + o(1)} \cdot l^{k - 1}\). This reduces the gap between the best lower and upper bounds from exponential to polynomial in k. We also generalise some of these results to the tournament setting.

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来源期刊
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
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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