On the Extremal Functions of Acyclic Forbidden 0-1 Matrices

The extremal theory of forbidden 0-1 matrices studies the asymptotic growth of the function Ex(P, n), which is the maximum weight of a matrix A is an element of{0, 1}(nxn) whose submatrices avoid a fixed pattern P is an element of{0, 1}(kxl). This theory has been wildly successful at resolving problems in combinatorics [Kla00, MT04, CK12], discrete and computational geometry [Fur90, Agg15, ES96, PS91, Mit92, BG91], structural graph theory [GM14, BGK(+)21, BKTW22] and the analysis of data structures [Pet10, KS20], particularly corollaries of the dynamic optimality conjecture [CGK(+)15b, CGK(+)15a, CGJ(+)23, CPY24]. All these applications use acyclic patterns, meaning that when P is regarded as the adjacency matrix of a bipartite graph, the graph is acyclic. The biggest open problem in this area is to bound ExpP, nq for acyclic P. Prior results [Pet11a, PS13] have only ruled out the strict O(n log n) bound conjectured by Furedi and Hajnal [FH92]. At the two extremes, it is consistent with prior results that for all P. Ex(P, n)<= n log(1+o(1)) n, and also consistent that for all epsilon > 0.there exists P. Ex(P, n) >= n(2-epsilon). In this paper we establish a stronger lower bound on the extremal functions of acyclic P. Specifically, for any t >= 1 we give a new construction of relatively dense 0-1 matrices with Theta(n(log n/log log n)(t)) 1s that avoid a certain acyclic pattern Xt. Pach and Tardos [PT06] have conjectured that this type of result is the best possible, i.e., no acyclic P exists for which Ex(P, n) >= n(log n)(omega(1)).