Relative Turán densities of ordered graphs

TL;DR

This work introduces the relative Turán density $\varrho(F)$ for ordered graphs, the largest fraction of edges preserved when extracting an $F$-free subgraph from any host graph, and compares it to the ordered Turán density $\pi_<(F)$. It proves that $\varrho(F)$ is invariant under blow-ups, i.e., $\varrho(F(t))=\varrho(F)$, and establishes a sharp value for monotone paths: $\varrho(P_k)=\frac{k-1}{2k}$ for all $k\ge2$, using a combination of quadratic geometry on the simplex and a family of structured graphs $G_\varepsilon(n,d)$ to control $P_k$-free subgraphs. The results extend the understanding of extremal behavior in ordered graphs, provide a concrete parallel to the classical undirected theory, and raise several open questions on cycles, classifications, and higher-order (hypergraph) analogues. The methods blend combinatorial constructions with probabilistic and analytic tools to achieve precise density bounds and invariance properties.

Abstract

We introduce a modification of the Turán density of ordered graphs and investigate this graph parameter.

Theorems & Definitions (10)

• Definition 1.1
• Theorem 1.2
• Proposition 1.3
• proof : Proof of Proposition \ref{['lem:1620']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof