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Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions $W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert W\rVert_\square=\sup\limits_{S,T\subseteq [0,1]}\Big|\int\limits_{S\times T}W(x,y)\,dx\,dy\Big|.$ As to this norm, we have the Counting Lemma, $\lVert W_1- W_2\rVert_\square=0$ implies the Hom density for any graph in them are equal: $t(H,W_1)=t(H, W_2).$

Is there any(may be weaker than cut norm) norm defined on $\mathcal{W}$ such that when $\lVert W\rVert=0$, the counting lemma holds: $t(H,W)=0.$

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  • $\begingroup$ What is "graphon"? What is Hom density? What are $t$ and $H$? $\endgroup$
    – LSpice
    Commented Jul 6 at 22:58

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