• Corpus ID: 239009443

Poset Ramsey numbers: large Boolean lattice versus a fixed poset

@inproceedings{Axenovich2021PosetRN,
  title={Poset Ramsey numbers: large Boolean lattice versus a fixed poset},
  author={Maria Axenovich and Christian Winter},
  year={2021}
}
Given partially ordered sets (posets) (P,≤P ) and (P ,≤P ′), we say that P ′ contains a copy of P if for some injective function f : P → P ′ and for any X,Y ∈ P , X ≤P Y if and only of f(X) ≤P ′ f(Y ). For any posets P and Q, the poset Ramsey number R(P,Q) is the least positive integer N such that no matter how the elements of an N -dimensional Boolean lattice are colored in blue and red, there is either a copy of P with all blue elements or a copy of Q with all red elements. We focus on a… 

Figures from this paper

References

SHOWING 1-10 OF 16 REFERENCES
Boolean Lattices: Ramsey Properties and Embeddings
TLDR
This work provides some general bounds on R(P, P′) and focuses on the situation when P and P′ are both Boolean lattices, giving asymptotically tight bounds for the number of copies of Qn in QN and for a multicolor version of a poset Ramsey number.
Ramsey Properties for $V$-shaped Posets in the Boolean Lattices
Given posets P1,P2, . . . ,Pk, let the Boolean Ramsey number R(P1,P2, . . . ,Pk) be the minimum number n such that no matter how we color the elements in the Boolean lattice Bn with k colors, there
Poset Ramsey Numbers for Boolean Lattices
A subposet $Q'$ of a poset $Q$ is a \textit{copy of a poset} $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x \le y$ in $P$ iff $f(x) \le f(y)$ in $Q'$. For posets $P,
The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets and Chains
TLDR
The exact values of the Boolean rainbow Ramsey number for general P and Q are determined, being the antichains, the Boolean posets, or the chains.
A Construction for Cube Ramsey
The (poset) cube Ramsey number R(Qn, Qn) is defined as the least m such that any 2-coloring of the m-dimensional cube Qm admits a monochromatic copy of Qn. The trivial lower bound R(Qn, Qn) ≥ 2n was
Forbidden induced subposets of given height
  • István Tomon
  • Computer Science, Mathematics
    J. Comb. Theory, Ser. A
  • 2019
TLDR
This paper shows that a special partition of the largest family of a partially ordered set can be used to derive bounds in a number of other extremal set theoretical problems and their generalizations in grids, such as the size of families avoiding weak posets, Boolean algebras, or two distinct sets and their union.
Ramsey Numbers for Partially-Ordered Sets
TLDR
A refinement of Ramsey numbers is presented by considering graphs with a partial ordering on their vertices, a natural extension of the ordered Ramsey numbers, and connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice are explored.
Rainbow Ramsey Problems for the Boolean Lattice
We address the following rainbow Ramsey problem: For posets P, Q what is the smallest number n such that any coloring of the elements of the Boolean lattice Bn either admits a monochromatic copy of P
Forbidden Hypermatrices Imply General Bounds on Induced Forbidden Subposet Problems
We prove that for every poset P, there is a constant CP such that the size of any family of subsets of {1, 2, . . ., n} that does not contain P as an induced subposet is at most
Set families with forbidden subposets
  • L. Lu, K. Milans
  • Computer Science, Mathematics
    J. Comb. Theory, Ser. A
  • 2015
TLDR
The Turan function of P, denoted $\pi^*(n,P)$, is the maximum size of a $P-free family of subsets of $\{1,\ldots,n\}$.
...
1
2
...