We investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies $x \vee \neg x = \top $ is no larger than $\frac {2}{3}$. Finally, we generalize our results to infinite Heyting algebras, and present their applications to point-set topology, black-box algebras, and the philosophy of logic.

What is the maximum number of copies of a fixed forest T in an n-vertex graph in a graph class $\mathcal {G}$ as $n\to \infty $ ? We answer this question for a variety of sparse graph classes $\mathcal {G}$ . In particular, we show that the answer is $\Theta (n^{\alpha _{d}(T)})$ where $\alpha _{d}(T)$ is the size of the largest stable set in the subforest of T induced by the vertices of degree at most d, for some integer d that depends on $\mathcal {G}$ . For example, when $\mathcal {G}$ is the class of k-degenerate graphs then $d=k$ ; when $\mathcal {G}$ is the class of graphs containing no $K_{s,t}$ -minor ( $t\geqslant s$ ) then $d=s-1$ ; and when $\mathcal {G}$ is the class of k-planar graphs then $d=2$ . All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.