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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, the Ramsey number R(G;H) is bounded below by (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. Together with Erdõs, he then asked when this lower bound is attained, introducing the notion of Ramsey goodness and its systematic study. We say that G is Hgood if the Ramsey number of (G;H) is equal to the general lower bound. Among other results, it was shown by Burr 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 3uniform hypergraphs. Motivated by Burr’s result concerning paths and a recent result of Balogh, Clemen, Skokan, and Wagner, we ask: what 3graphs H is a (long) tight path good for? We demonstrate that, in stark contrast to the graph case, long tight paths are generally not Hgood for various types of 3graphs H. Even more, we show that the ratio R(P_{n}, H)/n for a pair (P_{n}, H) consisting of a tight path on n vertices and a 3graph H cannot in general be bounded above by any function depending only on χ(H). We complement these negative results with a positive one, determining the Ramsey number asymptotically for pairs (P_{n}, H) when H belongs to a certain family of hypergraphs.
Our goal is to explore the notion of Ramsey goodness in the setting of 3uniform hypergraphs. Motivated by Burr’s result concerning paths and a recent result of Balogh, Clemen, Skokan, and Wagner, we ask: what 3graphs H is a (long) tight path good for? We demonstrate that, in stark contrast to the graph case, long tight paths are generally not Hgood for various types of 3graphs H. Even more, we show that the ratio R(P_{n}, H)/n for a pair (P_{n}, H) consisting of a tight path on n vertices and a 3graph H cannot in general be bounded above by any function depending only on χ(H). We complement these negative results with a positive one, determining the Ramsey number asymptotically for pairs (P_{n}, H) when H belongs to a certain family of hypergraphs.
Original language  English 

Title of host publication  EUROCOMB’23 
Publisher  Masaryk University Press 
Pages  17 
Number of pages  7 
DOIs  
Publication status  Published  28 Aug 2023 
Event  European Conference on Combinatorics, Graph Theory and Applications  Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic Duration: 28 Aug 2023 → 1 Sept 2023 https://iuuk.mff.cuni.cz/events/conferences/eurocomb23/ 
Publication series
Name  European Conference on Combinatorics, Graph Theory and Applications 

Publisher  Masaryk University Press 
Number  12 
ISSN (Electronic)  27883116 
Conference
Conference  European Conference on Combinatorics, Graph Theory and Applications 

Abbreviated title  EUROCOMB'23 
Country/Territory  Czech Republic 
City  Prague 
Period  28/08/23 → 1/09/23 
Internet address 
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