Let $G$ and $H$ be graphs, let $\vec H$ be a fixed orientation of $H$.

Denote by $D(G,\vec H)$ the number of orientations of $G$ that contain a copy of $\vec H$ and denote by $D'(G,H)$ the number of spanning subgraphs of $G$ that contain a copy of $H$.

Claim: $D'(G,H) \leq D(G, \vec H)$.

In [1] the claim is proven using some more general results involving set systems and shattering. Is there a simple, elementary (e.g. inductive) proof of the result?

[1] L.Kozma, S.Moran: Shattering, Graph Orientations, and Connectivity, 2012.

Let $G$ and $H$ be graphs, let $\vec H$ be a fixed orientation of $H$.

Denote by $D(G,\vec H)$ the number of orientations of $G$ that contain a copy of $\vec H$ and denote by $D'(G,H)$ the number of spanning subgraphs of $G$ that contain a copy of $H$.

Claim: $D'(G,H) \leq D(G, \vec H)$.

Note: a spanning subgraph of a graph $G=(V,E)$ is a graph $G_X=(V,X)$, where $X \subseteq E$.

In [1] the claim is proven using some more general results involving set systems and shattering. Is there a simple, elementary (e.g. inductive) proof of the result?

[1] L.Kozma, S.Moran: Shattering, Graph Orientations, and Connectivity, 2012.

Let $G$ and $H$ be graphs, let $\vec H$ be a fixed orientation of $H$.

Denote by $D(G,\vec H)$ the number of orientations of $G$ that contain a copy of $\vec H$ and denote by $D'(G,H)$ the number of subgraphs of $G$ (having the same vertex set) that contain a copy of $H$.

Claim: $D'(G,H) \leq D(G, \vec H)$.

In [1] the claim is proven using a more general theory of set systems and shattering. Is there a simple, elementary (e.g. inductive) proof of the result?

[1] L.Kozma, S.Moran: Shattering, Graph Orientations, and Connectivity, 2012.

Let $G$ and $H$ be graphs, let $\vec H$ be a fixed orientation of $H$.

Denote by $D(G,\vec H)$ the number of orientations of $G$ that contain a copy of $\vec H$ and denote by $D'(G,H)$ the number of spanning subgraphs of $G$ that contain a copy of $H$.

Claim: $D'(G,H) \leq D(G, \vec H)$.

In [1] the claim is proven using some more general results involving set systems and shattering. Is there a simple, elementary (e.g. inductive) proof of the result?

[1] L.Kozma, S.Moran: Shattering, Graph Orientations, and Connectivity, 2012.