VC-dimension of subsets of Hamming graphs

Abstract

Following recent work on the VC-dimension of subsets of various pseudorandom graphs, we study the VC-dimension of Hamming graphs, which have proved somewhat resistant to the standard techniques in the literature. Our methods are elementary, and agree with or improve upon previously known results. In particular, for $H(2,q)$ we show tight bounds on the size of a subset of vertices to guarantee VC-dimension 2 or 3. We also prove an assortment of results for other parameters, with many of these being tight as well.

Figures (5)

• Figure 1: The shattering set is $W=\{x,y\}.$ When two vertices are adjacent, an edge is indicated with a solid line. When two vertices may not be adjacent, this is indicated with a dotted curve. In the proof of Lemma \ref{['lemma22']}, $z$ served as $u{xy}.$
• Figure 2: The four cases, with the shattering points filled in, and other points of interest indicated with dotted lines.
• Figure 3: The vertices from the shattering set $W=\{x,y,z\}$ are circled. When two vertices are adjacent, an edge is indicated with a solid line. When two vertices may not be adjacent, this is indicated with a dotted curve. Note that as long as the edge relations are allowed, some vertices in the figure may be identified with one another. So, for example, $x$ cannot be $y,$ but it could be $u_{yz}.$
• Figure 4: Each $2\times 2\times 2$ subcube will either have five points (shaded) or zero (unshaded). Every layer will be a pair of copies of $H(2,q)$ will have $q/2$ of these subcubes, arranged diagonally. This pattern will shift with subsequent layer, so that shaded cubes are never in line with one another.
• Figure 5: This is a two-dimensional slice of $U_d"(12).$

Theorems & Definitions (31)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Proposition 1.5
• Proposition 1.6
• Proposition 1.7
• Theorem 1.8
• Lemma 2.1