# ComBench: A Benchmark for Rigorous Proof Reasoning and Constructive Realization in Olympiad-Level Combinatorics

arXiv: https://arxiv.org/abs/2606.10479 · HTML: https://mlsanity.com/paper/2606.10479/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 23 of its 33 references that are in the map.
- First posted on arXiv 9 Jun 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 23 of 23)

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- [FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI](https://mlsanity.com/paper/2411.04872/) (2024)
- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
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- [CombiBench: Benchmarking LLM Capability for Combinatorial Mathematics](https://mlsanity.com/paper/2505.03171/) (2025)
- [Reliable Fine-Grained Evaluation of Natural Language Math Proofs](https://mlsanity.com/paper/2510.13888/) (2025)
- [Measuring Progress in Reasoning Toward Mathematical Discovery with Automatic Verification](https://mlsanity.com/paper/2603.15617/) (2026)
- [Achieving Gold-Medal-Level Olympiad Reasoning via Simple and Unified Scaling](https://mlsanity.com/paper/2605.13301/) (2026)
- [Characterizing, Evaluating, and Optimizing Complex Reasoning](https://mlsanity.com/paper/2602.08498/) (2026)
- [Teaching Thinking Models to Reason with Tools: A Full-Pipeline Recipe for Tool-Integrated Reasoning](https://mlsanity.com/paper/2605.06326/) (2026)

## Abstract (arXiv)

Combinatorics is central to Olympiad-level mathematical problem solving, requiring deep discrete reasoning, creative constructions, and rigorous structural insight. Recent evidence suggests that even today's strongest frontier models remain uneven on Olympiad combinatorics, revealing a gap in creative mathematical reasoning. We introduce ComBench, an Olympiad-level combinatorics benchmark for evaluating and diagnosing the combinatorial reasoning capabilities of large language models. ComBench contains 100 human-annotated competition-level problems organized around two complementary settings: analysis-centric problems, which primarily require rigorous mathematical arguments, and construction-centric problems, which require explicit constructions in addition to correctness justifications. The evaluation protocol combines rubric-guided proof grading with deterministic construction verification, exposing cases where proof quality and construction validity diverge. Experiments on frontier open- and closed-source models show that ComBench is far from saturated: the strongest model reaches 65.4% overall Avg. and 75.3% overall Best@4. We further find that Rigorous Proof Reasoning and Constructive Realization are distinct capabilities: Kimi-K2.6 trails GPT-5.5 on analysis-centric proof grading but surpasses it on construction-centric Best@4, while Existence and Construction problems remain consistently hardest across representative frontier models.