Induced and non-induced poset saturation problems

@article{Keszegh2021InducedAN,
  title={Induced and non-induced poset saturation problems},
  author={Bal{\'a}zs Keszegh and Nathan Lemons and Ryan R. Martin and D{\"o}m{\"o}t{\"o}r P{\'a}lv{\"o}lgyi and Bal{\'a}zs Patk{\'o}s},
  journal={J. Comb. Theory, Ser. A},
  year={2021},
  volume={184},
  pages={105497}
}
View PDF on arXiv
11 Citations
Highly Influential Citations
1
Background Citations
3
Methods Citations
2
Results Citations
1

Tables from this paper

11 Citations

Citation Type

Citation Type

Almost all Permutation Matrices have Bounded Saturation Functions
TLDR
Fulek and Keszegh proved that for every 0-1 matrix $P, either $sat(n, P) = O(1)$ or $sat (n,P) = \Theta(n)$, and affirm their conjecture by proving that almost all permutation matrices $P$ have “almost all” $k \times k$ permutations matrices such thatsat( n, P).
C O ] 5 J ul 2 02 1 Induced and non-induced poset saturation problems
A subfamily G ⊆ F ⊆ 2[n] of sets is a non-induced (weak) copy of a poset P in F if there exists a bijection i : P → G such that p ≤P q implies i(p) ⊆ i(q). In the case where in addition p ≤P q holds…
Saturation Problems about Forbidden 0-1 Submatrices
TLDR
It is proved that every $0$-$1$ forbidden matrix has its saturation function either in $\Theta(1)$ or $\TheTA(n)$ in the case when the authors restrict ourselves to square saturating matrices.
Saturation problems in the Ramsey theory of graphs, posets and point sets
Continuous Tur\'an numbers
One of the most famous open problems in combinatorics is the Zarankiewicz problem, which asks for the maximum number of ones in an n × n matrix that has no s × t submatrix of all ones. The…
Forbidden subposet problems in the grid
Minimal Diamond-Saturated Families
For a given fixed poset P we say that a family of subsets of [n] is P-saturated if it does not contain an induced copy of P, but whenever we add to it a new set, an induced copy of P is formed. The…
Saturation for Small Antichains
For a given positive integer k we say that a family of subsets of [ n ] is k -antichain saturated if it does not contain k pairwise incomparable sets, but whenever we add to it a new set, we do find k…
Saturation of Ordered Graphs
Recently, the saturation problem of 0-1 matrices gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study…
Poset Ramsey numbers: large Boolean lattice versus a fixed poset
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…
...
...

References

SHOWING 1-10 OF 18 REFERENCES
Improved Bounds for Induced Poset Saturation
TLDR
For the 4-point poset known as the diamond, this work proves the minimum size of an induced-$\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\operatorname{sat}^*(n,\ mathcal{D}_2)\geq\sqrt{n}$, improving upon a logarithmic lower bound.
SATURATION FOR THE BUTTERFLY POSET
Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set…
On Saturated k-Sperner Systems
TLDR
The lower bound is shown that, if $|X|\geq k$ is sufficiently large with respect to $k, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements.
Saturating Sperner Families
TLDR
This paper addresses the problem of finding the minimum size of a family saturating the k-Sperner property and the minimumsized family that saturates the Spernerproperty and that consists only of l-sets and (l + 1)-sets.
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…
The saturation number of induced subposets of the Boolean lattice
Set Families with a Forbidden Subposet
  • B. Bukh
  • Mathematics
    Electron. J. Comb.
  • 2009
TLDR
This work asymptotically determines the size of the largest family F of subsets of subset of $\{1,\dots,n\}$ not containing a given poset P if the Hasse diagram of $P$ is a tree.
Progress on poset-free families of subsets
Increasing attention is being paid to the study of families of subsets of an n-set that contain no subposet P. Especially, we are interested in such families of maximum size given P and n. For…
On Families of Subsets With a Forbidden Subposet
TLDR
A new approach is used to apply methods from extremal graph theory and probability theory to identify new classes of posets H, for which La(n, H) can be determined asymptotically as n → ∞ for various poset H, including two-end-forks, up-down trees, and cycles C4k on two levels.
On a lemma of Littlewood and Offord
Remark. Choose Xi = l, n even. Then the interval ( — 1, + 1 ) contains Cn,m s u m s ^ i e ^ , which shows that our theorem is best possible. We clearly can assume that all the Xi are not less than 1.…
...
...