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Let $\mathbf{A}=\{A_i\}_{i=1}^{\infty}$ be a sequence of sets with each $A_i$being a non-empty collection of $0$-$1$ sequences of length $i$. For $x\in[0,1)$, the maximal run-length function $\ell_n(x,\mathbf{A})$ (with respect to$\mathbf{A}$) is defined to the largest $k$ such that in the first $n$ digitsof the dyadic expansion of $x$ there is a consecutive subsequence contained in$A_k$. Suppose that $\lim_{n\to\infty}(\log_2|A_n|)/n=\tau$ for some $\tau\in[0,1]$ and one additional assumption holds, we prove a generalization of theErd\H{o}s-R\'enyi limit theorem which states that\[\lim_{n\to\infty}\frac{\ell_n(x,\mathbf{A})}{\log_2n}=\frac{1}{1-\tau}\] forLebesgue almost all $x\in [0,1)$. For the exceptional sets, we prove under acertain stronger assumption on $\mathbf{A}$ that the set \[\left\{x\in [0,1):\lim_{n\to\infty}\frac{\ell_n(x,\mathbf{A})}{\log_2n}=0\text{ and }\lim_{n\to\infty}\ell_n(x,\mathbf{A})=\infty\right\}\] has Hausdorff dimensionat least $1-\tau$.

Maximal run-length function with constraints: a generalization of the Erd\H{o}s-R\'enyi limit theorem and the exceptional sets