[Paper Review] Extremal problems on ordered and convex geometric hypergraphs

Zoltán Füredi, Tao Jiang+3 more|arXiv (Cornell University)|Jul 13, 2018

Limits and Structures in Graph Theory7 references3 citations

TL;DR

This paper establishes extremal bounds for ordered and convex geometric $r$-uniform hypergraphs, introducing a splitting theorem that reveals a phase transition in extremal behavior based on codegree density. The key result is an exact extremal function for a specific ordered $r$-graph $F$ consisting of two alternating edges: ${ m{ex}}_{ ightarrow}(n,F) = inom{n}{r} - inom{n-r}{r}$, which serves as an ordered analog of the Erdős–Ko–Rado theorem.

ABSTRACT

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and discover a general partitioning phenomenon which allows us to determine the order of magnitude of the extremal function for various ordered and convex geometric hypergraphs. A special case is the ordered $n$-vertex $r$-graph $F$ consisting of two disjoint sets $e$ and $f$ whose vertices alternate in the ordering. We show that for all $n \geq 2r + 1$, the maximum number of edges in an ordered $n$-vertex $r$-graph not containing $F$ is exactly \[ {n \choose r} - {n - r \choose r}.\] This could be considered as an ordered version of the Erdős-Ko-Rado Theorem, and generalizes earlier results of Capoyleas and Pach and Aronov-Dujmovič-Morin-Ooms-da Silveira.

Motivation & Objective

• To extend extremal graph theory to ordered and convex geometric $r$-uniform hypergraphs, motivated by geometric and combinatorial applications.
• To resolve the ordered version of the Erdős–Ko–Rado theorem for a specific $r$-graph with alternating edges.
• To establish a general partitioning principle (splitting theorem) that relates extremal functions in ordered hypergraphs to those of bounded interval chromatic number.
• To investigate the difference in extremal behavior between linearly ordered and cyclically ordered hypergraphs, particularly for paths and matchings.
• To address open conjectures on linear or near-linear extremal functions for forests in ordered and convex geometric settings.

Proposed method

• Introduce the concept of a split hypergraph, where edges have two vertices in one interval and one in each of $r-1$ others, to model structured extremal configurations.
• Prove a splitting theorem showing that ${ m{ex}}_{ ightarrow}(n,F)$ is bounded by $O(z_{ ightarrow}(n,F) ext{ polylog }n)$ or $O(z_{ ightarrow}(n,F))$, depending on the codegree density $c$ of the forbidden hypergraph $F$.
• Construct extremal families $G(n,r,r+2)$ in both ordered and convex geometric settings to establish lower bounds for path and matching extremal functions.
• Use a recursive vertex selection argument to avoid forbidden configurations, particularly for crossing paths in the convex geometric setting.
• Leverage known results on convex geometric matchings (Capoyleas–Pach) to derive exact bounds for $CM_k^r$, showing ${ m{ex}}_{ ightarrow}(n,CM_k^r) = { m{ex}}_{igcirclearrowright}(n,CM_k^r)$ and $ heta(n^{r-1})$ growth.
• Apply extremal techniques from ordered and convex geometric graph theory to uniform hypergraphs, generalizing results on unit distances and crossing matchings.

Experimental results

Research questions

• RQ1 What is the exact extremal function for an ordered $r$-graph consisting of two disjoint edges with alternating vertices in the linear order?
• RQ2 How does the extremal function for ordered hypergraphs depend on the codegree density of the forbidden hypergraph?
• RQ3 Can the ordered Erdős–Ko–Rado theorem be generalized to $r$-uniform hypergraphs with alternating edge structures?
• RQ4 What is the extremal behavior of $r$-uniform crossing $k$-paths in the convex geometric setting compared to the ordered setting?
• RQ5 For which classes of $r$-uniform forests is the extremal function ${ m{ex}}_{ ightarrow}(n,F)$ bounded by $O(n^{r-1} ext{ polylog }n)$?

Key findings

• For all $n eq 2r+1$, the extremal function for the ordered $r$-graph $F$ with two alternating edges is exactly ${ m{ex}}_{ ightarrow}(n,F) = inom{n}{r} - inom{n-r}{r}$, providing an exact ordered analog of the Erdős–Ko–Rado theorem.
• The splitting theorem establishes that ${ m{ex}}_{ ightarrow}(n,F) = O(z_{ ightarrow}(n,F) ext{ polylog }n)$ when the forbidden hypergraph $F$ has codegree density $c = r-1$, and $O(z_{ ightarrow}(n,F))$ when $c > r-1$.
• For $r$-uniform crossing $k$-paths $CP_k^r$, the extremal function in the ordered setting satisfies ${ m{ex}}_{ ightarrow}(n,CP_k^r) = inom{n}{r} - inom{n-r}{r}$ for $k eq r+1$, and $c(r+1,r) = r$.
• In the convex geometric setting, the extremal function for $r$-uniform crossing $k$-matchings $CM_k^r$ satisfies ${ m{ex}}_{igcirclearrowright}(n,CM_k^r) = heta(n^{r-1})$, with no logarithmic factors.
• The extremal function for $CP_{2r}^r$ in the convex geometric setting is bounded above by $O(n^{r-1})$, and the construction $G(n,r,r+2)$ avoids such paths due to structural constraints on vertex ordering.
• The extremal function for $CM_k^r$ is identical in both ordered and convex geometric settings: ${ m{ex}}_{ ightarrow}(n,CM_k^r) = { m{ex}}_{igcirclearrowright}(n,CM_k^r)$, and this function is $ heta(n^{r-1})$.