On density of subgraphs of halved cubes

TL;DR

This paper extends the analysis of subgraph densities in hypercube-related graphs by introducing the clique-VC-dimension and applying shifting techniques to bound edge-to-vertex ratios in 1,2-inclusion graphs.

Contribution

It introduces the clique-VC-dimension and adapts shifting methods to bound densities of subgraphs of halved cubes, generalizing previous VC-dimension results.

Findings

Bound the ratio |E|/|V| to binomial coefficient based on clique-VC-dimension

Introduces the concept of clique-VC-dimension for set families

Shows that 1,2-inclusion graphs are subgraphs of halved cubes and Johnson graphs

Abstract

Let $\mathcal S$ be a family of subsets of a set $X$ of cardinality $m$ and $\text{VC-dim}(\mathcal S)$ be the Vapnik-Chervonenkis dimension of $\mathcal S$. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if $G_1(\mathcal S)=(V,E)$ is the subgraph of the hypercube $Q_m$ induced by $\mathcal S$ (called the 1-inclusion graph of $\mathcal S$), then $\frac{|E|}{|V|}\le \text{VC-dim}({\mathcal S})$. Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation. In this note, we adapt the shifting technique to prove that if $\mathcal S$ is an arbitrary set family and $G_{1,2}(\mathcal S)=(V,E)$ is the 1,2-inclusion graph of $\mathcal S$ (i.e., the subgraph of the square $Q^2_m$ of the hypercube $Q_m$ induced by $\mathcal S$), then $\frac{|E|}{|V|}\le \binom{d}{2}$, where $d:=\text{cVC-dim}^*(\mathcal S)$ is the clique-VC-dimension of $\mathcal S$ (which we introduce in this paper). The 1,2-inclusion graphs are exactly the subgraphs of halved cubes and comprise subgraphs of Johnson graphs as a subclass.