# An AI enhanced approach to the tree unimodality conjecture

arXiv: https://arxiv.org/abs/2510.18826 · HTML: https://mlsanity.com/paper/2510.18826/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 16 references that are in the map.
- First posted on arXiv 21 Oct 2025.

## Builds on (most cited first, 3 of 3)

- [PatternBoost: Constructions in Mathematics with a Little Help from AI](https://mlsanity.com/paper/2411.00566/) (2024)
- [Generative Modeling for Mathematical Discovery](https://mlsanity.com/paper/2503.11061/) (2025)
- [What makes math problems hard for reinforcement learning: a case study](https://mlsanity.com/paper/2408.15332/) (2024)

## Built on by (most cited first, 1 of 1)

- [Trees and Graphs with Non Log-concave Dominating Set Sequence via AI Tools](https://mlsanity.com/paper/2605.02193/) (2026)

## Abstract (arXiv)

Given a graph G, its independence sequence is the integral sequence a1,a2,...,an, where ai is the number of independent sets of vertices of size i. In the late 80's Alavi, Erdos, Malde, Schwenk showed that this sequence need not be unimodal for general graphs, but conjectured that it is always unimodal whenever G is a tree. This conjecture was then naturally generalized to claim that the independence sequence of trees should be log concave, in the sense that ai2 is always above ai-1ai+1. This conjecture stood for many years, until in 2023, Kadrawi, Levit, Yosef, and Mizrachi proved that there were exactly two trees on 26 vertices whose independence sequence was not log concave. In this paper, we use the AI architecture PatternBoost, developed by Charton, Ellenberg, Wagner, and Williamson to train a machine to find counter-examples to the log-concavity conjecture. We will discuss the successes of this approach - finding tens of thousands of new counter-examples to log-concavity with vertex set sizes varying from 27 to 101 - and some of its fascinating failures.