In Linear Hashing ($\mathsf{LH}$) with $\beta$ bins on a size $u$ universe ${\mathcal{U}=\{0,1,\ldots, u-1\}}$, items $\{x_1,x_2,\ldots, x_n\}\subset \mathcal{U}$ are placed in bins by the hash function $$x_i\mapsto (ax_i+b)\mod p \mod \beta$$ for some prime $p\in [u,2u]$ and randomly chosen integers $a,b \in [1,p]$. The "maxload" of $\mathsf{LH}$ is the number of items assigned to the fullest bin. Expected maxload for a worst-case set of items is a natural measure of how well $\mathsf{LH}$ distributes items amongst the bins. Fix $\beta=n$. Despite $\mathsf{LH}$'s simplicity, bounding $\mathsf{LH}$'s worst-case maxload is extremely challenging. It is well-known that on random inputs $\mathsf{LH}$ achieves maxload $\Omega\left(\frac{\log n}{\log\log n}\right)$; this is currently the best lower bound for $\mathsf{LH}$'s expected maxload. Recently Knudsen established an upper bound of $\widetilde{O}(n^{1 / 3})$. The question "Is the worst-case expected maxload of $\mathsf{LH}$ $n^{o(1)}$?" is one of the most basic open problems in discrete math. In this paper we propose a set of intermediate open questions to help researchers make progress on this problem. We establish the relationship between these intermediate open questions and make some partial progress on them.

翻译：在线性哈希（$\mathsf{LH}$）中，对于大小为$u$的全域${\mathcal{U}=\{0,1,\ldots, u-1\}}$及$\beta$个桶，元素$\{x_1,x_2,\ldots, x_n\}\subset \mathcal{U}$通过哈希函数$$x_i\mapsto (ax_i+b)\mod p \mod \beta$$分配到桶中，其中素数$p\in [u,2u]$，整数$a,b \in [1,p]$随机选取。$\mathsf{LH}$的"最大负载"指装载元素最多的桶中的元素数量。在最坏情况元素集下，最大负载的期望值是衡量$\mathsf{LH}$在桶间分布均匀性的自然指标。固定$\beta=n$。尽管$\mathsf{LH}$结构简单，但界定其最坏情况最大负载极具挑战性。已知在随机输入下$\mathsf{LH}$的最大负载为$\Omega\left(\frac{\log n}{\log\log n}\right)$，这是当前$\mathsf{LH}$期望最大负载的最佳下界。近期Knudsen给出了$\widetilde{O}(n^{1 / 3})$的上界。"$\mathsf{LH}$的最坏情况期望最大负载是否为$n^{o(1)}$？"是离散数学中最基本的未解决问题之一。本文提出一组中间开放问题以助研究者攻克此难题，并阐明了这些中间问题之间的关联，同时取得了部分阶段性进展。