A Survey of Forbidden Configuration Results

Abstract

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

Multi-symbol forbidden configurations

“…In [Ans13], there is a classification of asymptotic bounds of forb(m, F ) for (0, 1)-matrices F up to 5 rows. We wish to do the same for forb(m, r, Sym(F )) in this paper for simple (0, 1)-matrices F .…”

“…The classification of asymptotic bounds of forb(m, F ) for 5-rowed matrices F is still much an open problem. The current classification of asymptotic bounds of forb(m, F ) for 5-rowed matrices can be found in [Ans13].…”

Forbidden Subposet Problems for Traces of Set Families

“…One of them is the area of forbidden subconfigurations. If H ⊆ 2 [k] is a fixed family, then one can ask for the maximum size of a 'big' family F ⊆ 2 [n] such that for any k-subset X of [n], the trace F | X does not contain a subfamily isomorphic to H. For more details, the interested reader is referred to the survey of Anstee [2] and the references within. Naturally, one can consider several forbidden configurations at once.…”

Forbidden Families of Minimal Quadratic and Cubic Configurations

“…We solve all cases when the minimal simple cubic configuration has four rows. If Conjecture 8.1 of [3] is true, then there are no minimal simple cubic configurations on 5 rows. The six-rowed ones are discussed in Section 8.…”