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A hypergraph isomorphism is a bijection of the vertex sets such that a subset is a hyperedge of the first hypergraph if and only if its image is a hyperedge of the second. The condition preserves both edges and nonedges. In the two-uniform case it is the usual relation of isomorphic graphs.

A colouring is polychromatic with $r$ colours if every hypergraph edge contains a vertex of every colour. For $r\ge 2$, this is stronger than requiring every hypergraph edge to be nonmonochromatic. Under independent uniform colours, a $k$-element edge misses at least one of three colours with probability $3(2/3{)}^k-3(1/3{)}^k$. If each edge meets at most $m$ other edges, the edge-intersection dependency graph of events and the Lovász local lemma therefore give a polychromatic three-colouring whenever $e(m+1)[3(2/3{)}^k-3(1/3{)}^k]\le 1$.

An edge of a hypergraph is one of its specified subsets of the vertex set. In a uniform hypergraph all edges have the same cardinality; unlike an ordinary edge of a graph, an edge may contain more than two vertices.

A multipartite hypergraph has a specified partition of its vertex of a hypergraph set into parts, and every hypergraph edge meets each part in at most one vertex of a hypergraph. For an $r$-uniform hypergraph with $r$ parts, an hypergraph edge chooses exactly one vertex of a hypergraph from each part; its indicator is therefore a function on the Cartesian product of those parts. A quadripartite three-uniform hypergraph instead has four possible types of hypergraph edge, according to the omitted part. A choice of one vertex of a hypergraph from each of its four parts is a three-uniform tetrahedron exactly when all four triples are hypergraph edges. Specifying the parts makes both subset density and labelled counting precise.

For hypergraphs $F,H$, a labelled copy is an injective function $\varphi :V(F)\to V(H)$ taking every hypergraph edge of $F$ to an hypergraph edge of $H$. Specified vertex of a hypergraph classes may additionally require $\varphi (i)\in V_i$. The count is the sum, over these injective maps, of the product of the host hypergraph edge indicators. Without injectivity the same expression counts hypergraph homomorphisms. On a common host of size $n$, their counts differ by at most $\left(\genfrac{}{}{0ex}{}{|V(F)|}{2}\right)n^{|V(F)|-1}$: every noninjective map identifies at least one pair of abstract vertices of a hypergraph, each prescribed equality permits at most $n^{|V(F)|-1}$ maps, and the union bound completes the estimate. An induced copy additionally preserves nonedges, so its indicator product includes nonedge factors. The counting lemma for octahedrally quasirandom three-uniform hypergraphs estimates these labelled products when the requisite centered triple indicators have small box norm.

A vertex of a hypergraph $H=(V,E)$ is an element of its ground set $V$. Each hypergraph edge is a subset of $V$, and isolated vertices may belong to no edge. In a multipartite hypergraph, each vertex additionally lies in exactly one of the specified parts. This is the hypergraph analogue of a vertex of a graph, allowing edges of cardinality other than two.

A hypergraph is linear when two distinct hypergraph edges intersect in at most one vertex of a hypergraph. Equivalently, every pair of distinct vertices of a hypergraph has hypergraph codegree at most one. This restricts local overlap without bounding the hypergraph chromatic number: the random alteration method gives linear three-uniform hypergraphs of arbitrarily large hypergraph chromatic number.

A proper hypergraph colouring assigns colours to the vertices of a hypergraph so that no hypergraph edge is monochromatic. Each colour class is a hypergraph independent set. This is often called weak colouring, to distinguish it from requiring every pair of vertices in each edge to have different colours.

For a three-uniform hypergraph, the link of a vertex $v$ is the ordinary graph on the remaining vertices in which $xy$ is an edge precisely when $vxy$ is a hyperedge. Its edge count is the hypergraph vertex degree of $v$.

A matching in a hypergraph is a family of pairwise vertex-disjoint hypergraph edges. A perfect matching in a hypergraph covers every vertex; an almost-perfect matching in a hypergraph leaves a small proportion uncovered. In a proper edge colouring, each colour class is a matching in a hypergraph.

A subhypergraph restricts the vertex set of a hypergraph and retains some of its edges contained in that vertex set. Retaining every such edge gives an induced subhypergraph.

Two sign-vector vertices of a centered hypercube differing in $h$ coordinates have dot product $n-2h$. Thus their body diagonals can be orthogonal only in even dimension. A collection of mutually orthogonal nonzero diagonals has at most $n$ members; achieving $n$ is equivalent to the existence of a Hadamard matrix of order $n$. For unoriented lines the angle is determined by $|n-2h|/n$, while oriented vertex vectors retain the sign of their dot product.

An ideal hyperbolic triangle has all three vertices on the boundary at infinity of the hyperbolic plane. All its angles are zero and its hyperbolic area in curvature $-1$ is $\pi$, by hyperbolic triangle area.

For curvature minus one, no hyperbolic triangle contains an open hyperbolic disc of radius greater than $\frac{1}{2}\log 3$. Move the disc centre to zero in the Poincare disc model and enclose the triangle in an ideal triangle with the same radial vertex directions. One gap between ideal directions is at least $2\pi /3$. Its opposite geodesic is at distance at most $2\mathrm{artanh}(2-\sqrt{3})=\frac{1}{2}\log 3$ from zero. The symmetric ideal hyperbolic triangle attains the bound as its inradius.

The unit ball in Euclidean space with the displayed Riemannian metric models hyperbolic space of curvature minus one. Its ideal boundary is the unit sphere, and its geodesics are diameters or circle arcs orthogonal to that sphere. The metric is invariant under the conformal ball automorphisms.

An isometry of hyperbolic space with positive attained translation length has an invariant complete geodesic joining its two ideal fixed points. This is its axis. It translates along the axis by its translation length; in three dimensions it may additionally rotate around it. In upper half-space, $(z,t)\mapsto (kz,|k|t)$ with $|k|\ne 1$ has the vertical axis through zero and length $|\log |k||$. Every connecting path has length at least the change in $\log t$, and a vertical segment attains the bound.

For a nonempty finite set $\{p_1,\dots ,p_m\}$ in finite-dimensional hyperbolic space, the continuous function $R(x)={\max }_{j}d(x,p_j)$ attains its minimum. Indeed, a bounded candidate radius confines all possible improving centers to a closed bounded ball about $p_1$. Such a ball is compact in the Poincare ball model, since its points remain away from the ideal boundary. Minimizing $R$ on it supplies a closed ball of smallest enclosing radius. More generally this argument works for any nonempty bounded set in a proper metric space when the supremum-distance function is finite.

In the positive Hermitian determinant-one model of three-dimensional hyperbolic space, the base point is $I$ and a matrix $g\in SL(2,\mathbb{C})$ sends it to $g{g}^{\ast }$. The distance obeys $2\cosh d(I,g{g}^{\ast })=\mathrm{tr}(g{g}^{\ast })$. Bounded displacement therefore bounds every matrix entry, and the determinant-one condition gives a closed bounded subset of a finite-dimensional real vector space. Passing to $PSL(2,\mathbb{C})$ preserves compactness. Conjugating moves any chosen base point to $I$. A discrete subgroup meets each such compact set in finitely many elements, proving escape of distinct elements and proper discontinuity, including finite stabilizers.

An ideal hyperbolic quadrilateral is the region bounded by four hyperbolic geodesics whose four consecutive endpoints are distinct points on the ideal boundary. The ideal vertices lie outside the hyperbolic plane. In the hyperbolic disk, the four vertices $1,i,-1,-i$ give a regular example: its sides are the interior arcs of the circles of radius one centered at $\pm 1\pm i$. Although the sides have infinite length, the quadrilateral has finite hyperbolic area $2\pi$ in curvature $-1$.

A geodesic polygon in the hyperbolic plane is bounded by finitely many geodesic segments. For curvature $-1$, its area is its angle defect, as given by hyperbolic polygon area. Pairing equally long edges produces hyperbolic surfaces, with smoothness at glued vertices determined by their total angles.