scholarly journals On symmetric intersecting families of vectors

2021 ◽  
pp. 1-6
Author(s):  
Sean Eberhard ◽  
Jeff Kahn ◽  
Bhargav Narayanan ◽  
Sophie Spirkl
Keyword(s):  
Set Theory ◽  
Transitive Group ◽  
Threshold Behaviour ◽  
Extremal Set Theory ◽  
Sharp Threshold ◽  
The Third ◽  
Intersecting Families ◽  
Product Measures ◽  
Extremal Set ◽  
Group Of Symmetries

Abstract A family of vectors in [k] n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k] n invariant under a transitive group of symmetries is o(k n ), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.

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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