Davenport-Schinzel theory of matrices

On Nonlinear Forbidden 0–1 Matrices: A Refutation of a Füredi-Hajnal Conjecture

“…(Füredi-Hajnal [14]) If P is contained in P , where P, P are 0-1 matrices, then Ex(P , n) ≤ Ex(P, n). [14]) Let P ∈ {0, 1} k×l be a forbidden matrix where P (i, l − 1) = 1 (i.e., a 1 in the last column of P ) and let P ∈ {0, 1} k×(l+1) be identical to P in the first l columns and where [28]) Let P ∈ {0, 1} k×l be a forbidden matrix with a single 1 in the last column and let P ∈ {0, 1} k×(l−1) be P with the last column removed. Then Ex(P, n) = O(n + Ex(P , n) log n) and if Ex(P , n) = n 1+Ω(1) then Ex(P, n) = Θ(Ex(P , n))…”

“…Let H(P ) be the unordered (bipartite) graph corresponding to matrix P and let Ex T u (H, n) be the Turán-number of a graph H, i.e., the maximum number of edges in an n-vertex graph avoiding subgraphs isomorphic to H. At a high level the growth of Ex T u (H, n) is understood very well; it is Θ(n 2 ) if H is not bipartite, O(n) if H is a forest, and Ω(n 1+c1 ) and O(n 1+c2 ) in all other cases, for constants 0 < c 1 ≤ c 2 < 1 [7,13,22,2,6,5]. Füredi and Hajnal conjectured [14] that the extremal functions for 0-1 matrix avoidance and unordered subgraph avoidance never differ by more than a logarithmic factor:…”

“…Obviously if P is contained in P or is a stretched version of P , the nonlinearity of Ex(P , n) bears witness to the nonlinearity of Ex(P, n). For example, all nonlinear weight-4 matrices can be reduced to P 6 via zero or more stretching operations [14,33]. Since Ex(P 6 , n) = Θ(nα(n)) is nonlinear [14], it represents the sole cause of nonlinearity among weight-4 matrices.…”

“…Very recently the author [30] has shown that numerous data structures based on path compression and binary search trees can be analyzed in a simple, uniform way using the forbidden submatrix method. After the original applications of forbidden patterns P 1 and P 2 [11,4,27], Füredi and Hajnal [14] began a campaign to categorize all small forbidden patterns by their extremal function and to understand the prop-erties for forbidden patterns that influence their extremal functions. They made the important but simple observations that the forbidden submatrix framework essentially subsumes extremal problems in DavenportSchinzel sequences and Turán-type (unordered) subgraph avoidance.…”

“…In other words, the forbidden submatrix framework had actually been used for decades under different guises. Following [11,4], Füredi and Hajnal [14], and Tardos [33] managed to categorize the growth of Ex(P, n) for every weight-4 pattern P and Tardos [33] bounded the growth of Ex(P, n) for most sets P of weight-4 patterns. However, our current understanding of weight-5 and larger forbidden patterns is incomplete [33,28,10,17,15].…”