On the maximum density of a matrix and a transcendental Tur\'an-type density

TL;DR

This paper determines the maximum density of certain matrices and graphs, revealing a transcendental Turán-type density for a specific bipartite graph and providing explicit minimizers for matrix densities.

Contribution

It establishes the inducibility of P4 in ordered bipartite graphs, finds the asymptotic maximum density for matrices, and constructs explicit minimizers for all sizes.

Findings

Inducibility of P4 is 2/e^2 in ordered bipartite graphs.

Explicit minimizers are constructed for all h x h matrices.

Almost all h x h 0/1 matrices are minimizers as h grows.

Abstract

We prove that the inducibility of $P_4$ in ordered monotone balanced bipartite graphs is $2/e^2$, establishing the smallest known graph with transcendental Tur\'an-type density. Moreover, the limit object is a binary graphon, so it generates a deterministic model. This is a special case of a more general framework addressed here -- the asymptotic maximum density of a constant matrix over an arbitrary symbol set, in a large, possibly monotone, matrix. We solve all $2 \times 2$ monotone cases (one of which corresponds to the aforementioned $P_4$) and all but one of the $2 \times 2$ unrestricted cases. While $(h!/h^h)^2$ is a lower bound for the asymptotic maximum density of an $h \times h$ matrix, we explicitly construct, for all $h \ge 1$, an $h \times h$ minimizer, i.e., a matrix for which this bound is attained. We also sketch how known results on the inducibility of graphs can be modified to show that, as $h$ grows, almost all $h \times h$ $0/1$ matrices are minimizers.