Piercing all maximal cliques in hypergraphs
Ponente: Dániel Gábor Timon
Institución: Universidad Eötvös Lorand, Budapest, Hungría
Tipo de Evento: Investigación
In this work, I will talk about, we refuted this conjecture in a strong form. We show that for any constant c < 1 and integers k ≥ 3 and t ≥ 1, there exist k-uniform hypergraphs G whose maximum clique size exceeds c |V (G)|, yet the family of maximum size cliques of G cannot be pierced by t vertices. This demonstrates that no universal constant threshold guarantees bounded piercing number for maximum cliques in uniform hypergraphs.
We discuss further questions concerning the relationship between clique size and piercing maximum cliques in hypergraphs, and introduce a geometric variant of the problem using Helly’s Theorem.
Institución: Universidad Eötvös Lorand, Budapest, Hungría
Tipo de Evento: Investigación
| Cuándo |
06/05/2026 de 17:00 a 18:00 |
|---|---|
| Dónde | ZOOM ID 882 9372 3602 |
| Agregar evento al calendario |
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Graphs whose maximum clique size exceeds half of the total number of vertices satisfy a classical property: the family of their maximum sized cliques can be pierced by a single vertex. This result dates back to Hajnal. Motivated by his theorem, Jung, Keszegh, Pálvölgyi, and Yuditsky recently conjectured that an analogous result should hold for hypergraphs of larger uniformity, with an appropriate constant replacing the threshold 12.
In this work, I will talk about, we refuted this conjecture in a strong form. We show that for any constant c < 1 and integers k ≥ 3 and t ≥ 1, there exist k-uniform hypergraphs G whose maximum clique size exceeds c |V (G)|, yet the family of maximum size cliques of G cannot be pierced by t vertices. This demonstrates that no universal constant threshold guarantees bounded piercing number for maximum cliques in uniform hypergraphs.
We discuss further questions concerning the relationship between clique size and piercing maximum cliques in hypergraphs, and introduce a geometric variant of the problem using Helly’s Theorem.
