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arXiv · 2608.19414

The Non-Cancelling-Intersections Conjecture Fails for Left-Linear Trees

Hermann Wilhelm
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Abstract

First formulated by Amarilli, Monet, and Suciu (arXiv:2401.16210, 2024), the Non-Cancelling Intersections (NCI) conjecture is an open problem in combinatorics stating that any set union can be constructively built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In the same paper, two orthogonal possible strengthenings are proposed: using only left-linear trees, and using non-trivial intersections only positively or only negatively depending on the sign of their M\"obius value. Here we show that using only left-linear trees, the conjecture is false (independent of the other strengthening). Our argument is non-constructive. We prove the existence of a counterexample, though it is of immense size.

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Hermann Wilhelm. 2026-08-19. The Non-Cancelling-Intersections Conjecture Fails for Left-Linear Trees. https://arxiv.org/abs/2608.19414

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