AI 中文总结
该研究证实了Alon-Spencer关于上升波的猜想，即大小≥n/2的集合中上升波长度的下界可移除log log n因子。
AI 中文摘要
对于正整数n，记[n]={1,…,n}。严格递增序列x₁<⋯<xₖ为上升波，若其相邻差非递减。设g(n)为满足：所有大小≥n/2的集合A⊆[n]都包含长度为k的上升波的最大整数k。Alon与Spencer证明，对所有足够大的n，有c₁(log n)²/log log n ≤ g(n) ≤ c₂(log n)²，并猜想下界中的log log n因子可移除；本文证实了该猜想。
英文摘要
For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. A strictly increasing sequence of integers $x_1<\cdots<x_k$ is an \emph{ascending wave} if its consecutive differences are nondecreasing. Let $g(n)$ be the largest integer $k$ such that every set $A\subseteq[n]$ with $|A|\ge n/2$ contains an ascending wave of length $k$. Alon and Spencer proved that \[ c_1\frac{(\log n)^2}{\log\log n}\le g(n)\le c_2(\log n)^2 \] for all sufficiently large $n$, and they conjectured that the factor $\log\log n$ in the lower bound can be removed. In this paper, we confirm their conjecture.
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