Martin Balko, Mate Vizer

Research output: Contribution to journalArticlepeer-review


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 .

Original languageEnglish
Pages (from-to)214-228
Number of pages15
JournalSIAM Journal on Discrete Mathematics
Issue number1
StatePublished - 1 Jan 2022
Externally publishedYes


  • multipartite hypergraphs
  • ordered Ramsey numbers
  • ordered hypergraphs

ASJC Scopus subject areas

  • Mathematics (all)


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