When Joints Meet Extremal Graph Theory: Hypergraph Joints

TL;DR

This paper extends the joints problem in incidence geometry to hypergraphs, establishing bounds on hypergraph joints and generalizing classical combinatorial theorems like Kruskal--Katona and Friedgut--Kahn.

Contribution

It introduces hypergraph joints and proves a generalized joints theorem, connecting geometric incidence problems with hypergraph combinatorics, extending prior bounds to a broader context.

Findings

Established an upper bound on the number of hypergraph joints.

Generalized the partial shadow phenomenon to all hypergraphs.

Connected the results to classical inequalities like H"older's inequality.

Abstract

The Kruskal--Katona theorem determines the maximum number of $d$-cliques in an $n$-edge $(d-1)$-uniform hypergraph. A generalization of the theorem was proposed by Bollob\'as and Eccles, called the partial shadow problem. The problem asks to determine the maximum number of $r$-sets of vertices that contain at least $d$ edges in an $n$-edge $(d-1)$-uniform hypergraph. In our previous work, we obtained an asymptotically tight upper bound via its connection to the joints problem, a problem in incidence geometry. In a different direction, Friedgut and Kahn generalized the Kruskal--Katona theorem by determining the maximum number of copies of any fixed hypergraph in an $n$-edge hypergraph, up to a multiplicative factor. In this paper, using the connection to the joints problem again, we generalize our previous work to show an analogous partial shadow phenomenon for any hypergraph, generalizing Friedgut and Kahn's result. The key idea is to encode the graph-theoretic problem with new kinds of joints that we call hypergraph joints. Our main theorem is a generalization of the joints theorem that upper bounds the number of hypergraph joints, which the partial shadow phenomenon immediately follows from. In addition, with an appropriate notion of multiplicities, our theorem also generalizes a generalization of H\"older's inequality considered by Finner.