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 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.

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 Small Forbidden Configurations II

10.37236/1548 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Richard Anstee ◽  
Ron Ferguson ◽  
Attila Sali
Keyword(s):  
Graph Theory ◽  
Directed Graphs ◽  
Column Permutation ◽  
Vertex Ordering ◽  
Forbidden Configuration ◽  
Forbidden Configurations ◽  
Exact Bounds

The present paper continues the work begun by Anstee, Griggs and Sali on small forbidden configurations. In the notation of (0,1)-matrices, we consider a (0,1)-matrix $F$ (the forbidden configuration), an $m\times n$ (0,1)-matrix $A$ with no repeated columns which has no submatrix which is a row and column permutation of $F$, and seek bounds on $n$ in terms of $m$ and $F$. We give new exact bounds for some $2\times l$ forbidden configurations and some asymptotically exact bounds for some other $2\times l$ forbidden configurations. We frequently employ graph theory and in one case develop a new vertex ordering for directed graphs that generalizes Rédei's Theorem for Tournaments. One can now imagine that exact bounds could be available for all $2\times l$ forbidden configurations. Some progress is reported for $3\times l$ forbidden configurations. These bounds are improvements of the general bounds obtained by Sauer, Perles and Shelah, Vapnik and Chervonenkis.

scholarly journals Small Forbidden Configurations III

10.37236/997 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
R. P. Anstee ◽  
N. Kamoosi
Keyword(s):  
Column Permutation ◽  
Simple Matrix ◽  
Forbidden Configuration ◽  
Forbidden Configurations ◽  
Q Matrix

The present paper continues the work begun by Anstee, Ferguson, Griggs and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let $F$ be a $k\times l$ (0,1)-matrix (the forbidden configuration). Assume $A$ is an $m\times n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define ${\hbox{forb}}(m,F)$ as the largest $n$, which would depend on $m$ and $F$, so that such an $A$ exists. 'Small' refers to the size of $k$ and in this paper $k=2$. For $p\le q$, we set $F_{pq}$ to be the $2\times (p+q)$ matrix with $p$ $\bigl[{1\atop0}\bigr]$'s and $q$ $\bigl[{0\atop1}\bigr]$'s. We give new exact values: ${\hbox{forb}}(m,F_{0,4})=\lfloor {5m\over2}\rfloor +2$, ${\hbox{forb}}(m,F_{1,4})=\lfloor {11m\over4}\rfloor +1$, ${\hbox{forb}}(m,F_{1,5})=\lfloor {15m\over4}\rfloor +1$, ${\hbox{forb}}(m,F_{2,4})=\lfloor {10m\over3}-{4\over3}\rfloor$ and ${\hbox{forb}}(m,F_{2,5})=4m$ (For ${\hbox{forb}}(m,F_{1,4})$, ${\hbox{forb}}(m,F_{1,5})$ we obtain equality only for certain classes modulo 4). In addition we provide a surprising construction which shows ${\hbox{forb}}(m,F_{pq})\ge \bigl({p+q\over2}+O(1)\bigr)m$.

scholarly journals A Survey of Forbidden Configuration Results

10.37236/2379 ◽  
2013 ◽  
Vol 1000 ◽  
Author(s):  
Richard Anstee
Keyword(s):  
Fundamental Result ◽  
Asymptotic Results ◽  
Column Permutation ◽  
The Matrix ◽  
Simple Matrix ◽  
Forbidden Configuration

Let $F$ be a $k\times \ell$ (0,1)-matrix. We say a (0,1)-matrix $A$ has $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a trace and in the language of hypergraphs a configuration is a subhypergraph.Let $F$ be a given $k\times \ell$ (0,1)-matrix. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. The matrix $F$ need not be simple. We define $\hbox{forb}(m,F)$ as the maximum number of columns of any simple $m$-rowed matrix $A$ which do not contain $F$ as a configuration. Thus if $A$ is an $m\times n$ simple matrix which has no submatrix which is a row and column permutation of $F$ then $n\le\hbox{forb}(m,F)$. Or alternatively if $A$ is an $m\times (\hbox{forb}(m,F)+1)$ simple matrix then $A$ has a submatrix which is a row and column permutation of $F$. We call $F$ a forbidden configuration. The fundamental result is due to Sauer, Perles and Shelah, Vapnik and Chervonenkis. For $K_k$ denoting the $k\times 2^k$ submatrix of all (0,1)-columns on $k$ rows, then $\hbox{forb}(m,K_k)=\binom{m}{k-1}+\binom{m}{k-2}+\cdots \binom{m}{0}$. We seek asymptotic results for $\hbox{forb}(m,F)$ for a fixed $F$ and as $m$ tends to infinity . A conjecture of Anstee and Sali predicts the asymptotically best constructions from which to derive the asymptotics of $\hbox{forb}(m,F)$. The conjecture has helped guide the research and has been verified for $k\times \ell$ $F$ with $k=1,2,3$ and for simple $F$ with $k=4$ as well as other cases including $\ell=1,2$. We also seek exact values for $\hbox{forb}(m,F)$. 

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 Genetic Algorithms Applied to Problems of Forbidden Configurations

10.37236/717 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
R. P. Anstee ◽  
Miguel Raggi
Keyword(s):  
Genetic Algorithm ◽  
Genetic Algorithms ◽  
Local Search ◽  
Search Algorithms ◽  
Local Search Algorithms ◽  
Column Permutation ◽  
Simple Matrix ◽  
Forbidden Configurations

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say a (0,1)-matrix $A$ avoids $F$ (as a configuration) if there is no submatrix of $A$ which is a row and column permutation of $F$. Let $\|{A}\|$ denote the number of columns of $A$. We define $\mathrm{forb}(m,F)=\max\{\|{A}\|\ : A$ is an $m$-rowed simple matrix that avoids $F \}$. Define an extremal matrix as an $m$-rowed simple matrix $A$ with that avoids $F$ and $\|{A}\|=\mathrm{forb}(m,F)$. We describe the use of Local Search Algorithms (in particular a Genetic Algorithm) for finding extremal matrices. We apply this technique to two forbidden configurations in turn, obtaining a guess for the structure of an $m\times\mathrm{forb}(m,F)$ simple matrix avoiding $F$ and then proving the guess is indeed correct. The Genetic Algorithm was also helpful in finding the proof.

scholarly journals Large forbidden configurations and design theory

2016 ◽  
Vol 53 (2) ◽  
pp. 157-166
Author(s):  
R. P. Anstee ◽  
Attila Sali
Keyword(s):  
Lower Bounds ◽  
Design Theory ◽  
Upper Bounds ◽  
Column Permutation ◽  
Forbidden Configurations

Let forb(m, F) denote the maximum number of columns possible in a (0, 1)-matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F. We consider cases where the configuration F has a number of columns that grows with m. For a k × l matrix G, define s · G to be the concatenation of s copies of G. In a number of cases we determine forb(m, mα · G) is Θ(mk+α). Results of Keevash on the existence of designs provide constructions that can be used to give asymptotic lower bounds. An induction idea of Anstee and Lu is useful in obtaining upper bounds.

scholarly journals Forbidden Berge Hypergraphs

10.37236/6482 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
R. P. Anstee ◽  
Santiago Salazar
Keyword(s):  
Extremal Function ◽  
Column Permutation ◽  
Vertex Graph ◽  
Simple Matrix ◽  
The Extremal Function

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a Berge hypergraph if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $\|A\|$ denote the number of columns in $A$, we define the extremal function $\mathrm{Bh}(m,{ F})=\max\{\|A\|\,:\, A \hbox{ }m\hbox{-rowed simple matrix and no Berge hypergraph }F\}$. We determine the asymptotics of $\mathrm{Bh}(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determining the maximum number of copies of $K_r$ in a $m$-vertex graph that has no $K_{s,t}$ subgraph, a problem studied by Alon and Shikhelman.

scholarly journals Forbidden Families of Minimal Quadratic and Cubic Configurations

10.37236/6902 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Attila Sali ◽  
Sam Spiro
Keyword(s):  
Extremal Function ◽  
Asymptotic Growth ◽  
Column Permutation ◽  
Simple Matrix ◽  
The Extremal Function

A matrix is simple if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a configuration, denoted $F\prec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $\mathcal{F}$ be a family of matrices. We define the extremal function $\text{forb}(m, \mathcal{F}) = \max\{|A|\colon A \text{ is an }m-\text{rowed simple matrix and has no configuration } F\in\mathcal{F}\}$. We consider pairs $\mathcal{F}=\{F_1,F_2\}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $\text{forb}(m, F_i)$ has greater asymptotic growth than $\text{forb}(m, \mathcal{F})$, extending research started by Anstee and Koch.

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