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Among families of a given size in the Boolean hypercube, a binary initial segment maximizes the number of contained two-dimensional cube faces. Under section compression in binary order, faces crossing the selected coordinate correspond to internal edges in the intersection of its two sections. The edge extremal theorem and mathematical induction control that contribution and the faces within the sections.

If the members of a possibly infinite set family $\mathcal{F}$ have size at most $r$, there is a finite subset ${\mathcal{F}}_0\subseteq \mathcal{F}$ of size at most $M(r,s)$ with exactly the same hitting sets of size at most $s$. Use mathematical induction on $s$. For nonempty $\mathcal{F}$, choose $A_0\in \mathcal{F}$. When $s=0$ this one member suffices. Otherwise, for each $x\in A_0$, retain an inductive kernel for ${\mathcal{F}}_x=\{A\in \mathcal{F}:x\notin A\}$ at parameter $s-1$, and take their set union together with $A_0$. Its size is at most $1+rM(r,s-1)=M(r,s)$. A hitting set $H$ for this kernel meets $A_0$ at some $x$; $H\setminus \{x\}$ meets the kernel for ${\mathcal{F}}_x$, hence all of ${\mathcal{F}}_x$, while $x$ meets the other members of $\mathcal{F}$. If $A_0$ is empty no hitting set exists, and if $\mathcal{F}$ is empty its kernel is empty. Only finite branching is used, so no finiteness assumption on $\mathcal{F}$ is needed.

For positive integers $k,s,\ell$, some $N$ ensures that every $k$-color finite coloring of $[N]$ contains a monochromatic setIt follows from the Van der Waerden theorem by mathematical induction on the number of colors. If $M$ is a bound for $k-1$ colors, take a long one-color arithmetic progression of length $(\ell -1)M+1$ and step $d$. Either one of $sd,2sd,\dots ,Msd$ has its color, giving the result immediately, or these multiples use only $k-1$ colors. Pull back their finite coloring to $[M]$ and use mathematical induction, then multiply the resulting configuration by $sd$. Enlarge the ambient finite integer interval to include all these multiples. The cases $k=1$ and $\ell =1$ are immediate. The parameter $s$ makes this useful for positive monochromatic solutions of a partition regular equation with arbitrary integer coefficients.

A directed system of first-order structures and coherent elementary embeddings has a direct limit into which every member embeds elementarily. Represent an element by a pair consisting of a stage and an element at that stage; identify two representatives when their images agree at a common later stage. Interpret functions and relations at a common stage. Mathematical induction on first-order formulas proves elementarity: for an existential first-order formula true at the limit, its witness and parameters occur at a common stage, and elementarity pulls truth back to the stage containing the original parameters. This generalizes the elementary chain theorem beyond linearly ordered diagrams.

Extend a finite coloring of the positive integers to a point $x$ of a two-sided full shift. A minimal subsystem of its forward orbit closure, together with the proximal-minimal existence theorem, supplies a minimal point $y$ proximal to $x$. Put $q=y(0)$. If all sums in $\{0\}\cup \mathrm{FS}(a_1,\dots ,a_r)$, the augmented finite-sums set, have color $q$ in $y$, their coordinate constraints define a cylinder set containing $y$. The joint return lemma for a proximal minimal pair chooses a new positive term so that all new sums have color $q$ in both $x$ and $y$. Mathematical induction gives an infinite monochromatic finite-sums set. The new term can exceed the sum of all previous terms, giving unique representations. The proximal-minimal existence theorem is a substantive input to this proof.

For an atomic formula $P$ (including logical equality), put $P^N=\neg \neg P$, and put ${\perp }^N=\perp$. Preserve logical conjunction, logical implication and universal quantification recursively, while settingLogical negation is logical implication to logical falsity, so $(\neg A{)}^N=\neg A^N$. The displayed logical disjunction and existence clauses are intuitionistically equivalent to the usual negative forms $\neg (\neg A^N\wedge \neg B^N)$ and $\neg \forall x\neg A^N$. Every translated first-order formula is stable by stability of a formula under double negation. In classical logic the translation is equivalent to the original first-order formula by mathematical induction and double-negation elimination.

If $A$ is a finite $K$-approximate group, thenfor every positive integer $m$, where $A^2\subseteq XA$ and $|X|\le K$. This follows immediately by mathematical induction.

For nonnegative integers $n,m$, the binomial coefficients satisfyThe identity follows by mathematical induction: adding the next term combines two adjacent binomial coefficients through Pascal's identity. An equivalent form is ${\sum }_{r=j}^N\left(\genfrac{}{}{0ex}{}{r}{j}\right)=\left(\genfrac{}{}{0ex}{}{N+1}{j+1}\right)$, a diagonal sum in Pascal's triangle. It provides closed forms for sums of binomial coefficients without expanding individual factorials.

If $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf, projection to the last coordinate makes any coherent $\mathcal{F}\subseteq {\mathcal{O}}_X^r$ an extension of a coherent submodule of ${\mathcal{O}}_X^{r-1}$ by an ideal sheaf. The long exact sequence in sheaf cohomology proves the assertion by mathematical induction on $r$.

If $n\ge 2d+1$ and the field characteristic is zero or greater than $d+1$, the matrix recording inclusion of $d$-sets in $(d+1)$-sets has full row rank. Subtracting equations on sets differing in one coordinate reduces the kernel problem to degree $d-1$ on $n-2$ coordinates. Mathematical induction then makes all coefficients equal, and the factor $d+1$ forces them to vanish. This gives the auxiliary-polynomial independence used in the uniform Frankl-Wilson theorem.

If total functions $g$ and $h$ are represented by lambda terms $G,H$, defineandAfter $j$ iterations the Church pair holds $(c_j,c_{f(\mathbf{x},j)})$, proved by mathematical induction using the defining recursion equations. Consequently $R$ represents the total primitive recursive function $f$. This statement concerns total inputs and total recursion constituents; it is not an unguarded composition theorem for arbitrary partial functions.

For a nonempty $r$-uniform set family, define $x>r-1$ by $|\mathcal{F}|=\left(\genfrac{}{}{0ex}{}{x}{r}\right)$ using real binomial coefficients. Then its lower shadow has at least $\left(\genfrac{}{}{0ex}{}{x}{r-1}\right)$ members. This continuous estimate is often easier to use than the exact integer expansion in the Kruskal-Katona theorem.

A short proof uses coordinate shifts of a set family. In a fully left-shifted set family split at coordinate one into ${\mathcal{F}}_0$ and ${\mathcal{F}}_1$, deleting one in the present section. The shifting property implies $\partial {\mathcal{F}}_0\subseteq {\mathcal{F}}_1$, and the total lower shadow has size $|{\mathcal{F}}_1|+|\partial {\mathcal{F}}_1|$. Induct on uniformity and, at fixed uniformity, on set family size. If $|{\mathcal{F}}_1|<\left(\genfrac{}{}{0ex}{}{x-1}{r-1}\right)$, the Pascal's identity and the size mathematical induction force $|\partial {\mathcal{F}}_0|>\left(\genfrac{}{}{0ex}{}{x-1}{r-1}\right)$, a contradiction. The uniformity mathematical induction applied to ${\mathcal{F}}_1$, followed by the Pascal's identity, proves the estimate. Integer $x$ is sharp for the full $r$-level on $x$ coordinates.