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The standard methodology of evaluating large language models (LLMs) based on static pairs of inputs and outputs is insufficient for developing assistants: this kind of assessments fails to take into account the essential interactive element in their deployment, and therefore limits how we understand language model capabilities. We introduce CheckMate, an adaptable prototype platform for humans to interact with and evaluate LLMs. We conduct a study with CheckMate to evaluate three language models~(InstructGPT, ChatGPT, and GPT-4) as assistants in proving undergraduate-level mathematics, with a mixed cohort of participants from undergraduate students to professors of mathematics. We release the resulting interaction and rating dataset, MathConverse. By analysing MathConverse, we derive a preliminary taxonomy of human behaviours and uncover that despite a generally positive correlation, there are notable instances of divergence between correctness and perceived helpfulness in LLM generations, amongst other findings. Further, we identify useful scenarios and existing issues of GPT-4 in mathematical reasoning through a series of case studies contributed by expert mathematicians. We conclude with actionable takeaways for ML practitioners and mathematicians: models which communicate uncertainty, respond well to user corrections, are more interpretable and concise may constitute better assistants; interactive evaluation is a promising way to continually navigate the capability of these models; humans should be aware of language models' algebraic fallibility, and for that reason discern where they should be used.

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random neighbour $y$ of $x$, then the probability that $y\in A$ is at least $\eta$. The work of this paper was motivated by an attempt to obtain a qualitative description of closed subsets of the Cayley graph $\Gamma$ whose vertex set is $\mathbb F_2^{n_1}\otimes \dots \otimes \mathbb F_2^{n_d}$ with two vertices joined by an edge if their difference is of the form $u_1\otimes \cdots \otimes u_d$. For the matrix case (that is, when $d=2$), such a description was obtained by Khot, Minzer and Safra, a breakthrough that completed the proof of the 2-to-2 conjecture. In this paper, we formulate a conjecture for higher dimensions, and prove it in an important special case. Also, we identify a statement about $\eta$-closed sets in Cayley graphs on arbitrary finite Abelian groups that implies the conjecture and can be considered as a "highly asymmetric Balog-Szemer\'edi-Gowers theorem" when it holds. We conclude the paper by showing that this statement is not true for an arbitrary Cayley graph. It remains to decide whether the statement can be proved for the Cayley graph $\Gamma$.