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arXiv · 2503.15608

Strict Erd\H{o}s-Ko-Rado theorems for simplicial complexes

Denys Bulavka·Russ Woodroofe
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Abstract

We show that if a simplicial complex is a near-cone of sufficiently high depth, then the only maximum families of small pairwise intersecting faces are those with a common intersection. Thus, near-cones of sufficiently high depth satisfy the strict Erd\H{o}s-Ko-Rado property conjectured by Holroyd and Talbot and by Borg. One consequence is a strict Erd\H{o}s-Ko-Rado theorem for independence complexes of chordal graphs with an isolated vertex. Under stronger shiftedness conditions, we prove a sharper stability theorem of Hilton-Milner type, as well as two cross-intersecting theorems.

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Denys Bulavka, Russ Woodroofe. 2025-03-19. Strict Erd\H{o}s-Ko-Rado theorems for simplicial complexes. https://arxiv.org/abs/2503.15608

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