Gerke–Marciniszyn–Steger sparse counting lemma

Let $H$ be a fixed graph. For each vertex $x\in V(H)$, let $V_x$ be a pairwise disjoint set of $n$ vertices, and let $\mathcal G(H,n,m)$ be the family of graphs with vertex set $\bigcup_{x\in V(H)}V_x$ and exactly $m$ edges between $V_x$ and $V_y$ for every $\{x,y\}\in E(H)$. Let $\mathcal G(H,n,m,\varepsilon)$ be the subfamily in which every such bipartite graph is $\varepsilon$-regular. Define

$$m_2(H)=\max\left\{\frac{|E(H')|-1}{|V(H')|-2}:H'\subseteq H,\ |V(H')|\geq 3\right\}.$$

Let $\mathcal F(H,n,m,\delta)$ consist of graphs containing fewer than

$$(1-\delta)n^{|V(H)|}\left(\frac{m}{n^2}\right)^{|E(H)|}$$

canonical copies of $H$. Counting Lemma. For any $\beta>0$ and $\delta>0$, there exist constants $\varepsilon>0$, $C>0$, and $n_0>0$ such that for all $m\geq Cn^{2-1/m_2(H)}$ and $n\geq n_0$,

$$|\mathcal F(H,n,m,\delta)\cap\mathcal G(H,n,m,\varepsilon)|\leq\beta^m {n^2\choose m}^{|E(H)|}.$$

The sparse counting lemma would provide quantitative control on the number of copies of a fixed graph in sparse regular configurations, complementing the resolved sparse embedding lemma (the KŁR conjecture). Its general form remains widely open.