An Improvement of the General Bound on the Largest Family of Subsets Avoiding a Subposet

@article{Grsz2017AnIO,
  title={An Improvement of the General Bound on the Largest Family of Subsets Avoiding a Subposet},
  author={D{\'a}niel Gr{\'o}sz and Abhishek Methuku and Casey Tompkins},
  journal={Order},
  year={2017},
  volume={34},
  pages={113-125}
}
Let La(n, P) be the maximum size of a family of subsets of [n] = {1, 2, … , n} not containing P as a (weak) subposet, and let h(P) be the length of a longest chain in P. The best known upper bound for La(n, P) in terms of |P| and h(P) is due to Chen and Li, who showed that La(n,P)≤1m+1|P|+12(m2+3m−2)(h(P)−1)−1n⌊n/2⌋$\text {La}(n,P) \le \frac {1}{m+1} \left (|{P}| + \frac {1}{2}(m^{2} +3m-2)(h(P)-1) -1 \right ) {\left (\begin {array}{c}{n}\\ {\lfloor n/2 \rfloor } \end {array}\right )}$ for any… Expand
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References

SHOWING 1-10 OF 17 REFERENCES
A Note on the Largest Size of Families of Sets with a Forbidden Poset
TLDR
An improved upper bound is provided on the size of families of subsets of [n] that do no contain a given poset P as a subposet that can be any positive integer less than $\lceil \frac{n}{2}\rceil$. Expand
On crown-free families of subsets
  • L. Lu
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 2014
TLDR
This paper proves that La ( n, O 2 t ) = ( 1 + o ( 1 ) ) ( n ⌊ n 2 ⌋ ) for all odd t ≥ 7 and that this is true for all even t ≥ 4. Expand
Poset-free families and Lubell-boundedness
Given a finite poset P, we consider the largest size La ( n , P ) of a family F of subsets of n ] : = { 1 , ? , n } that contains no subposet P. This continues the study of the asymptotic growth ofExpand
Diamond-free families
TLDR
It is conjectured that @p(P):=lim"n"->"~La(n,P)/(n@?n2@?) exists for general posets P, and, moreover, it is an integer. Expand
No four subsets forming an N
TLDR
Borders are given on how large F can be such that no four distinct sets A,B,C,D@?F satisfy A@?B, C@? B, C @?D, and the maximum size satisfies (n@?n2@?)(1+1n+@W(1n^2), which is very similar to the best-known bounds for the more restrictive problem of F avoiding three sets B,D,D. Expand
Induced and Non-induced Forbidden Subposet Problems
TLDR
The asymptotic behavior of La(n,P) is determined, the maximum size that an induced P-free subposet of the Boolean lattice B_n can have for the case when P is the complete two-level poset or the complete multi-level Poset K_{r,s,t} when all $s_i$'s either equal 4 or are large enough and satisfy an extra condition. Expand
Largest Families Without an r-Fork
TLDR
The maximum size of a finite set, F, is asymptotically determined up to the second term, improving the result of Tran. Expand
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 mostExpand
The method of double chains for largest families with excluded subposets
TLDR
A new method is introduced, counting the intersections of $\mathcal{F}$ with double chains, rather than chains, to prove the theorems of La(n,P) for infinitely many P posets. Expand
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.Expand
...
1
2
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