AI 中文总结
研究整数随机子集中k项等差数列构建，玩家Builder在预算b限制下逐个决定是否选择随机整数，确定了t = ω(n^{1 - 2/k})时，预算b = Θ((n/t)^((k - 2)/2))对Builder以高概率成功构建k项等差数列是必要且充分的最优阈值。
AI 中文摘要
2025年，Frieze、Krivelevich和Michaeli引入了随机图过程的受限预算版本，研究只能购买有限数量随机边的在线玩家构建结构的问题。本文将此框架从随机图转移到整数随机子集，聚焦于k项等差数列的构建。玩家Builder会看到从[n]中均匀随机抽取的t个整数序列，随着元素逐个揭示，Builder必须立即且不可撤销地决定是否选择当前整数，总选择元素的最大预算为b。我们确定了此过程的最优阈值，证明对于t = ω(n^{1 - 2/k})，预算b = Θ((n/t)^((k - 2)/2))对于Builder以高概率成功构建k项等差数列既是必要的也是充分的。
英文摘要
A restricted-budget version of the random graph process, introduced by Frieze, Krivelevich, and Michaeli in 2025, studies the construction of structures by an online player who can purchase only a limited number of random edges. In this paper, we transfer this framework from random graphs to random subsets of integers, focusing on the construction of $k$-term arithmetic progressions. A player, Builder, is presented with a sequence of $t$ integers drawn uniformly at random from $[n]$. As the elements are revealed one by one, Builder must immediately and irrevocably decide whether to select the current integer, subject to a maximum budget of $b$ selected elements in total. We establish the optimal thresholds for this process, proving that for $t = ω(n^{1-2/k})$, a budget of $b = Θ((n/t)^{\frac{k-2}{2}})$ is both necessary and sufficient for Builder to successfully construct a $k$-term arithmetic progression with high probability.
