scholarly journals A Generalization of the Bollobás Set Pairs Inequality

10.37236/9627 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Jason O'Neill ◽  
Jacques Verstraete
Keyword(s):  
Set Theory ◽  
Maximum Size ◽  
Covering Problem ◽  
Fundamental Result ◽  
Extremal Set Theory ◽  
Natural Connection ◽  
Extremal Set

The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for $n \geqslant k \geqslant t \geqslant 2$, we consider a collection of $k$ families $\mathcal{A}_i: 1 \leq i \leqslant k$ where $\mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \}$ so that $A_{1, i_1} \cap \cdots \cap A_{k,i_k} \neq \varnothing$ if and only if there are at least $t$ distinct indices $i_1,i_2,\dots,i_k$. Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size $\beta_{k,t}(n)$ of the families with ground set $[n]$.

scholarly journals Sperner's Theorem and a Problem of Erdős, Katona and Kleitman

2014 ◽  
Vol 24 (4) ◽  
pp. 585-608 ◽  
Author(s):  
SHAGNIK DAS ◽  
WENYING GAN ◽  
BENNY SUDAKOV
Keyword(s):  
Set Theory ◽  
Middle Level ◽  
Extremal Set Theory ◽  
Paper Making ◽  
Sperner's Theorem ◽  
Central Result ◽  
Extremal Set

A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain, F1 ⊂ F2. Erdős extended this theorem to determine the largest family without a k-chain, F1 ⊂ F2 ⊂ . . . ⊂ Fk. Erdős and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds.In 1966, Kleitman resolved this question for 2-chains, showing that the number of such chains is minimized by taking sets as close to the middle level as possible. Moreover, he conjectured the extremal families were the same for k-chains, for all k. In this paper, making the first progress on this problem, we verify Kleitman's conjecture for the families whose size is at most the size of the k + 1 middle levels. We also characterize all extremal configurations.

scholarly journals Disjoint Genus-0 Surfaces in Extremal Graph Theory and Set Theory Lead To a Novel Topological Theorem

2021 ◽  
Author(s):  
Arturo Tozzi
Keyword(s):  
Graph Theory ◽  
Set Theory ◽  
Extremal Graph Theory ◽  
Topological Manifold ◽  
Tree Graph ◽  
Extremal Set Theory ◽  
Extremal Set ◽  
Cycle Graph ◽  
Topological Theorem ◽  
Theory Lead

When an edge is removed, a cycle graph Cn becomes a n-1 tree graph. This observation from extremal set theory leads us to the realm of set theory, in which a topological manifold of genus-1 turns out to be of genus-0. Starting from these premises, we prove a theorem suggesting that a manifold with disjoint points must be of genus-0, while a manifold of genus-1 cannot encompass disjoint points.

scholarly journals Freiman's Theorem in Finite Fields via Extremal Set Theory

2009 ◽  
Vol 18 (3) ◽  
pp. 335-355 ◽  
Author(s):  
BEN GREEN ◽  
TERENCE TAO
Keyword(s):  
Set Theory ◽  
Finite Fields ◽  
Common Theme ◽  
Additive Combinatorics ◽  
Extremal Set Theory ◽  
Extremal Set ◽  
Freiman’S Theorem ◽  
Sharp Version

Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asymptotically sharp version of Freiman's theorem in $\F_2^n$: if $A \subseteq \F_2^n$ is a set for which |A + A| ≤ K|A| then A is contained in a subspace of size $2^{2K + O(\sqrt{K}\log K)}|A|$; except for the $O(\sqrt{K} \log K)$ error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K46) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman–Ruzsa conjecture in this case. A common theme in the arguments is the use of compression techniques. These have long been familiar in extremal set theory, but have been used only rarely in the additive combinatorics literature.

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