On the maximum density of a matrix and a transcendental Turán-type density

TL;DR

The paper develops a general framework for Turán-type density questions in matrices, defining $d(H,M)$, the asymptotic maximum $f(H)$, and a monotone variant $f^*(H)$. It proves a transcendental Turán-type density in the matrix setting by establishing that the inducibility of $P_4$ in ordered monotone balanced bipartite graphs equals $2/e^2$, with the limit object being a binary graphon, thus yielding a deterministic model. It further shows that for every order $h$ there exist explicit $h\times h$ minimizers achieving the universal lower bound $(h!)^2/h^{2h}$ and that almost all $0/1$ matrices are minimizers, using a combination of blowup constructions and probabilistic/variational arguments. These results extend inducibility and Turán-type density phenomena from graphs to matrices, reveal a natural transcendental density in this setting, and provide concrete minimizers and a probabilistic perspective on their typicality. The work highlights the interplay between exact constructions, variational calculus, and limit objects (binary graphons) in extremal matrix theory, with potential implications for deterministic limit models and beyond.

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 Turán-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.

Figures (5)

• Figure 1: The $2 \times 2$ matrices, up to density isomorphism.
• Figure 2: Hasse diagram of refinement for $2 \times 2$ matrices.
• Figure 3: The limit object of a sequence of matrices whose density of $\left(0001\right)$ approaches $2/e^2$. To comply with matrix notation, the vertical axis corresponds to the row index fraction from top to bottom and the horizontal axis corresponds to column index fraction from left to right. The shaded area represents the symbol $1$ and the non-shaded area represents the symbol $0$.
• Figure 4: Some elements of ${\mathcal{F}}_{2,3}^\sigma$ where $\sigma=(0,1)$.
• Figure 5: Depicted are a $\sigma$-type and three $\sigma$-flags $F,F' \in {\mathcal{F}}_{2,2}^\sigma$, $K=(M,\{2\},\{2\}) \in {\mathcal{F}}_{3,4}^\sigma$. Notice that $K$ has $\binom{2}{1}\binom{3}{1}=6$$2 \times 2$ subflags. Only two of them equal $F$, so $p(F,K)=\frac{1}{3}$. Similarly, $K$ has $\binom{2}{1}\binom{3}{1}\binom{1}{1}\binom{2}{1}=12$ pairs of $2 \times 2$ subflags such that the unmarked rows in both are disjoint and the unmarked columns in both are disjoint. Only one such pair corresponds to $(F,F')$. The first, corresponding to $F$, is given by choosing rows $\{1,2\}$ and columns $\{2,3\}$ of $M$, and the second, corresponding to $F'$, is given by choosing rows $\{2,3\}$ and columns $\{1,2\}$ of $M$. Hence, $p(F,F';K)=\frac{1}{12}$.

Theorems & Definitions (19)

• Conjecture 1.1
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Corollary 1.6
• Theorem 1.7
• Conjecture 1.8
• Lemma 2.1
• proof
• Lemma 2.2