We study the problem of selecting a statement that finds common ground across diverse population preferences. Generative AI is uniquely suited for this task because it can access a practically infinite set of statements, but AI systems like the Habermas machine leave the choice of generated statement to a voting rule. What it means for this rule to find common ground, however, is not well-defined. In this work, we propose a formal model for finding common ground in the infinite alternative setting based on the proportional veto core from social choice. To provide guarantees relative to these infinitely many alternatives and a large population, we wish to satisfy a notion of proportional veto core using only query access to the unknown distribution of alternatives and voters. We design an efficient sampling-based algorithm that returns an alternative in the (approximate) proportional veto core with high probability and prove matching lower bounds, which show that no algorithm can do the same using fewer queries. On a synthetic dataset of preferences over text, we confirm the effectiveness of our sampling-based algorithm and compare other social choice methods as well as LLM-based methods in terms of how reliably they produce statements in the proportional veto core.