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TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
A Survey of Forbidden Configuration Results
, 2010
"... This paper surveys various results concerning forbidden configurations (also known as trace). Let F be a k×ℓ (0,1)matrix (the forbidden configuration). 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. Considering the collection of al ..."
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This paper surveys various results concerning forbidden configurations (also known as trace). Let F be a k×ℓ (0,1)matrix (the forbidden configuration). 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. Considering the collection of all mrowed A which do not contain F as a configuration, we define forb(m, F) as the maximum number of columns of any matrix in the collection. Thus if A is an m × n simple matrix which has no submatrix which is a row and column permutation of F then n ≤ forb(m, F). Or alternatively if A is an m×(forb(m, F)+1) simple matrix then A has a submatrix which is a row and column permutation of F. Perhaps the most fundamental result is due to Sauer, Perles and Shelah, Vapnik and Chervonenkis but there are many other results catalogued here. We seek exact values for forb(m, F) as well as seeking asymptotic results for 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 forb(m, F). This conjecture has helped guide the research and has been verified for k ×ℓ F with k = 1, 2, 3, with ℓ = 1, 2 and for simple F with k = 4 as well as other cases.
Small Forbidden Configurations II
, 2001
"... 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 \Theta n (0,1)matrix A with no repeated columns which has no submatrix which is a row and colu ..."
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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 \Theta 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 \Theta l forbidden configurations and some asymptotically exact bounds for some other 2 \Theta l forbidden configurations. We frequently employ graph theory and in one case develop a new vertex ordering for directed graphs that generalizes R'edei's Theorem for Tournaments. One can now imagine that exact bounds could be available for all 2 \Theta l forbidden configurations. Some progress is reported for 3 \Theta l forbidden configurations. These bounds are improvements of the general bounds obtained by Sauer, Perles and Shelah, Vapnik and Chervonenkis.