Abstract
We introduce the following simpler variant of the Turán problem: Given integers n> k> r≥ 2 and m≥ 1 , what is the smallest integer t for which there exists an r -uniform hypergraph with n vertices, t edges and m connected components such that any k -subset of the vertex set contains at least one edge? We prove some general estimates for this quantity and for its limit, normalized by (nr) , as n→ ∞ . Moreover, we give a complete solution of the problem for the particular case when k= 5 , r= 3 and m≥ 2 .
Original language | English |
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Pages (from-to) | 168-182 |
Number of pages | 15 |
Journal | Acta Mathematica Hungarica |
Volume | 170 |
Issue number | 1 |
Early online date | 25 May 2023 |
DOIs | |
Publication status | Published - Jun 2023 |
Bibliographical note
Funding Information:Raffaella Mulas was supported by the Max Planck Society’s Minerva Grant.
Publisher Copyright:
© 2023, Akadémiai Kiadó, Budapest, Hungary.
Funding
Raffaella Mulas was supported by the Max Planck Society’s Minerva Grant.
Keywords
- hypergraph
- Turán number
- Turán problem