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Abstract
Given a pair of kuniform hypergraphs (G,H), the Ramsey number of (G,H), denoted by R(G,H), is the smallest integer n such that in every red/bluecolouring of the edges of K_{n}^{(k)} there exists a red copy of G or a blue copy of ~H. Burr showed that, for any pair of graphs (G,H), where G is large and connected, R(G,H)≥(v(G)−1)(χ(H)−1)+σ(H), where σ(H) stands for the minimum size of a colour class over all proper χ(H)colourings of H. We say that G is Hgood if R(G,H) is equal to the general lower bound. Burr showed that, for any graph ~H, every sufficiently long path is Hgood.
Our goal is to explore the notion of Ramsey goodness in the setting of kuniform hypergraphs. We demonstrate that, in stark contrast to the graph case, kuniform ℓpaths are not Hgood for a large class of kgraphs. On the other hand, we prove that long loose paths are always at least asymptotically Hgood for every H and derive lower and upper bounds that are best possible in a certain sense.
In the 3uniform setting, we complement our negative result with a positive one, in which we determine the Ramsey number asymptotically for pairs containing a long tight path and a 3graph H when H belongs to a certain family of hypergraphs. This extends a result of Balogh, Clemen, Skokan, and Wagner for the Fano plane asymptotically to a much larger family of 3graphs.
Our goal is to explore the notion of Ramsey goodness in the setting of kuniform hypergraphs. We demonstrate that, in stark contrast to the graph case, kuniform ℓpaths are not Hgood for a large class of kgraphs. On the other hand, we prove that long loose paths are always at least asymptotically Hgood for every H and derive lower and upper bounds that are best possible in a certain sense.
In the 3uniform setting, we complement our negative result with a positive one, in which we determine the Ramsey number asymptotically for pairs containing a long tight path and a 3graph H when H belongs to a certain family of hypergraphs. This extends a result of Balogh, Clemen, Skokan, and Wagner for the Fano plane asymptotically to a much larger family of 3graphs.
Original language  English 

Article number  104021 
Number of pages  24 
Journal  European Journal of Combinatorics 
Early online date  20 Jul 2024 
DOIs  
Publication status  Epub ahead of print  20 Jul 2024 
Bibliographical note
Not yet published as of 17/07/2024.Fingerprint
Dive into the research topics of 'Ramsey goodness of kuniform paths, or the lack thereof'. Together they form a unique fingerprint.
Ramsey theory: an extremal perspective
Engineering & Physical Science Research Council
1/01/22 → 31/12/24
Project: Research Councils

Matchings and tilings in graphs
Engineering & Physical Science Research Council
1/03/21 → 29/02/24
Project: Research Councils