Forbidden configurations and repeated induction

Anstee, Richard P.; Meehan, Connor

doi:10.1016/j.disc.2011.07.005

Cited by 2 publications

(3 citation statements)

References 17 publications

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“…We offer no insight on the exact value of m k . Theorem 3 is somewhat anticipated in [5] for the case t = 1 (Theorems 1.3,1.4 in [5]), but the more general results were handicapped searching for base cases which the proof here has avoided. The columns we add to K k are t copies of the…”

Section: This Doesn't Directly Solve [K 4 |T • K Tmentioning

confidence: 98%

“…Theorem 4 is anticipated in Theorem 1.7 in [5] for the case t = 1 but progress in [5] was stopped by difficulty with base cases.…”

Section: Extensions Of 2 • K K With the Same Boundmentioning

confidence: 99%

“…The goal of this paper is to show that there are many columns you can add to K k without changing the bound. Initial investigations (Theorem 1.2 in [5]) were hampered by trying to prove base cases. We use stability results to overcome this.…”

Section: Introductionmentioning

confidence: 99%

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Shattering and More: Extending the Complete Object

Anstee

Nikov

2022

Electron. J. Combin.

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Let $\mathcal{F}\subseteq 2^{[m]}$ be a family of subsets of $[m]=\{1,2,\ldots,m\}$. For $S\subseteq [m]$, let $\mathcal{F}|_S$ be the trace $\mathcal{F}|_S=\{B\cap S : B\in\mathcal{F}$, considered as a multiset. We say $\mathcal{F}$ shatters a set $S\subseteq [m]$ if $\mathcal{F}|_S$ has all $2^{|S|}$ possible sets (i.e. complete). We say $\mathcal{F}$ has a shattered set of size $k$ if $\mathcal{F}$ shatters some $S\subseteq [m]$ with $|S|=k$. It is well known that if $\mathcal{F}$ has no shattered $k$-set then $|\mathcal{F}|\le\binom{m}{k-1}+\binom{m}{k-2}+\cdots+\binom{m}{0}$. We obtain the same exact bound when forbidding less. Namely, given fixed positive integers $t$ and $k$, for every set $S\subseteq [m]$ with $|S|=k$, $\mathcal{F}$ is such that $\mathcal{F}|_S$ does not have both all possible sets $2^S$ and specified additional sets occuring at least $t$. Similar results are proven for double shattering, namely when $\mathcal{F}|_S$ does not have all sets $2^|S|$ appearing twice. The paper is written in matrix notation with trace replaced by configuration.

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Section: This Doesn't Directly Solve [K 4 |T • K Tmentioning

confidence: 98%

“…Theorem 4 is anticipated in Theorem 1.7 in [5] for the case t = 1 but progress in [5] was stopped by difficulty with base cases.…”

Section: Extensions Of 2 • K K With the Same Boundmentioning

confidence: 99%

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Shattering and More: Extending the Complete Object

Anstee

Nikov

2022

Electron. J. Combin.

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A Survey of Forbidden Configuration Results

Anstee

2013

Electron. J. Combin.

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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 $\hbox{forb}(m,F)$.

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Forbidden configurations and repeated induction

Cited by 2 publications

References 17 publications

Shattering and More: Extending the Complete Object

Shattering and More: Extending the Complete Object

A Survey of Forbidden Configuration Results

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