scholarly journals Exponential multivalued forbidden configurations

2021 ◽  
Vol vol. 23 no. 1 (Combinatorics) ◽  
Author(s):  
Travis Dillon ◽  
Attila Sali
Keyword(s):  
Set Theory ◽  
Stability Result ◽  
Identity Matrix ◽  
Extremal Set Theory ◽  
Column Permutation ◽  
Forbidden Configurations ◽  
Exact Bounds ◽  
Extremal Set ◽  
The Way

The forbidden number $\mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in $\{0,1,\dots,r-1\}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2\times 2$ identity matrix. Along the way, we expose some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$. Comment: 12 pages; v3: formatted for DMTCS; v2: Corollary 3.2 added, typos fixed, some proofs clarified

scholarly journals Forbidden Configurations: Exact Bounds Determined by Critical Substructures

10.37236/322 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
R. P. Anstee ◽  
S. N. Karp
Keyword(s):  
Set Theory ◽  
Theory Problem ◽  
Extremal Set Theory ◽  
Column Permutation ◽  
Simple Matrix ◽  
Forbidden Configurations ◽  
Exact Bounds ◽  
Extremal Set

We consider the following extremal set theory problem. Define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. An $m$-rowed simple matrix corresponds to a family of subsets of $\{1,2,\ldots ,m\}$. Let $m$ be a given integer and $F$ be a given (0,1)-matrix (not necessarily simple). We say a matrix $A$ has $F$ as a configuration if a submatrix of $A$ is a row and column permutation of $F$. We define $\hbox{forb}(m,F)$ as the maximum number of columns that a simple $m$-rowed matrix $A$ can have subject to the condition that $A$ has no configuration $F$. We compute exact values for $\hbox{forb}(m,F)$ for some choices of $F$ and in doing so handle all $3\times 3$ and some $k\times 2$ (0,1)-matrices $F$. Often $\hbox{forb}(m,F)$ is determined by $\hbox{forb}(m,F')$ for some configuration $F'$ contained in $F$ and in that situation, with $F'$ being minimal, we call $F'$ a critical substructure.

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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