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 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

Small forbidden configurations V: Exact bounds for 4 × 2 cases

2011 ◽  
Vol 48 (1) ◽  
pp. 1-22
Author(s):  
Richard Anstee ◽  
Farzin Barekat ◽  
Attila Sali
Keyword(s):  
Column Permutation ◽  
Single Column ◽  
Simple Matrix ◽  
Forbidden Configuration ◽  
Forbidden Configurations ◽  
Exact Bounds

The present paper continues the work begun by Anstee, Ferguson, Griggs, Kamoosi and Sali on small forbidden configurations. We define a matrix to besimpleif it is a (0, 1)-matrix with no repeated columns. LetFbe ak× (0, 1)-matrix (the forbidden configuration). AssumeAis anm×nsimple matrix which has no submatrix which is a row and column permutation ofF. We define forb (m, F) as the largestn, which would depend onmandF, so that such anAexists.DefineFabcdas the (a+b+c+d) × 2 matrix consisting ofarows of [11],brows of [10],crows of [01] anddrows of [00]. With the exception ofF2110, we compute forb (m; Fabcd) for all 4 × 2Fabcd. A number of cases follow easily from previous results and general observations. A number follow by clever inductions based on a single column such as forb (m; F1111) = 4m− 4 and forb (m; F1210) = forb (m; F1201) = forb (m; F0310) = (2m)+m+ 2 (proofs are different). A different idea proves forb (m; F0220) = (2m) + 2m− 1 with the forbidden configuration being related to a result of Kleitman. Our results suggest that determining forb (m; F2110) is heavily related to designs and we offer some constructions of matrices avoidingF2110using existing designs.

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 Color critical hypergraphs and forbidden configurations

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Richard Anstee ◽  
Balin Fleming ◽  
Zoltán Füredi ◽  
Attila Sali
Keyword(s):  
Linear Algebra ◽  
Upper Bound ◽  
Sufficient Condition ◽  
Column Permutation ◽  
International Audience ◽  
Simple Matrix ◽  
Forbidden Configuration ◽  
Forbidden Configurations

International audience The present paper connects sharpenings of Sauer's bound on forbidden configurations with color critical hypergraphs. We define a matrix to be \emphsimple if it is a $(0,1)-matrix$ with no repeated columns. Let $F$ be $a k× l (0,1)-matrix$ (the forbidden configuration). Assume $A$ is an $m× n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define $forb(m,F)$ as the best possible upper bound on n, for such a matrix $A$, which depends on m and $F$. It is known that $forb(m,F)=O(m^k)$ for any $F$, and Sauer's bond states that $forb(m,F)=O(m^k-1)$ fore simple $F$. We give sufficient condition for non-simple $F$ to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.

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