TY - JOUR

T1 - ON ORDERED RAMSEY NUMBERS of TRIPARTITE 3-UNIFORM HYPERGRAPHS

AU - Balko, Martin

AU - Vizer, Mate

N1 - Funding Information:
\ast Received by the editors March 15, 2021; accepted for publication (in revised form) September 13, 2021; published electronically January 10, 2022. An extended abstract version of this paper appeared in the Proceedings of Eurocomb 2021; see [4]. https://doi.org/10.1137/21M1404958 Funding: This article is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (grant 810115). The first author was supported by grant 18-13685Y of the Czech Science Foundation (GACR), and by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004). The second author was supported by the Hungarian National Research, Development and Innovation Office - NKFIH under grants SNN 129364, KH 130371, and FK 132060, by the J\a'nos Bolyai Research Fellowship of the Hungarian Academy of Sciences, and by the New National Excellence Program under grant UNKP-20-5-BME-45.\'
Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics

PY - 2022/1/1

Y1 - 2022/1/1

N2 - For an integer k \geq 2, an ordered k-uniform hypergraph \scrH = (H, <) is a k-uniform hypergraph H together with a fixed linear ordering < of its vertex set. The ordered Ramsey number R(\scrH, \scrG ) of two ordered k-uniform hypergraphs \scrH and \scrG is the smallest N \in \BbbN such that every red-blue coloring of the hyperedges of the ordered complete k-uniform hypergraph \scrK N(k) on N vertices contains a blue copy of \scrH or a red copy of \scrG . The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs. In particular, we prove that for all d, n \in \BbbN and for every ordered 3-uniform hypergraph \scrH on n vertices with maximum degree d and with interval chromatic number 3 there is an \varepsilon = \varepsilon (d) > 0 such that R(\scrH, \scrH ) \leq 2O(n2 - \varepsilon ). In fact, we prove this upper bound for the number R(\scrG, \scrK 3(3)(n)), where \scrG is an ordered 3-uniform hypergraph with n vertices and maximum degree d, and \scrK 3(3)(n) is the ordered complete tripartite hypergraph with consecutive color classes of size n. We show that this bound is not far from the truth by proving R(\scrH, \scrK 3(3)(n)) \geq 2\Omega (n log n) for some fixed ordered 3-uniform hypergraph \scrH .

AB - For an integer k \geq 2, an ordered k-uniform hypergraph \scrH = (H, <) is a k-uniform hypergraph H together with a fixed linear ordering < of its vertex set. The ordered Ramsey number R(\scrH, \scrG ) of two ordered k-uniform hypergraphs \scrH and \scrG is the smallest N \in \BbbN such that every red-blue coloring of the hyperedges of the ordered complete k-uniform hypergraph \scrK N(k) on N vertices contains a blue copy of \scrH or a red copy of \scrG . The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs. In particular, we prove that for all d, n \in \BbbN and for every ordered 3-uniform hypergraph \scrH on n vertices with maximum degree d and with interval chromatic number 3 there is an \varepsilon = \varepsilon (d) > 0 such that R(\scrH, \scrH ) \leq 2O(n2 - \varepsilon ). In fact, we prove this upper bound for the number R(\scrG, \scrK 3(3)(n)), where \scrG is an ordered 3-uniform hypergraph with n vertices and maximum degree d, and \scrK 3(3)(n) is the ordered complete tripartite hypergraph with consecutive color classes of size n. We show that this bound is not far from the truth by proving R(\scrH, \scrK 3(3)(n)) \geq 2\Omega (n log n) for some fixed ordered 3-uniform hypergraph \scrH .

KW - multipartite hypergraphs

KW - ordered Ramsey numbers

KW - ordered hypergraphs

UR - http://www.scopus.com/inward/record.url?scp=85130609169&partnerID=8YFLogxK

U2 - 10.1137/21M1404958

DO - 10.1137/21M1404958

M3 - Article

AN - SCOPUS:85130609169

SN - 0895-4801

VL - 36

SP - 214

EP - 228

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -