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The balanced complete multipartite blow-up $K_r(t)$ replaces each vertex of the complete graph $K_r$ by an independent set of $t$ vertices, placing every possible edge between different sets and none inside a set. Every fixed graph of chromatic number at most $r$ embeds into $K_r(t)$ for sufficiently large $t$.

For a finite simple graph, $\alpha (G)\ge {\sum }_{v\in V(G)}1/(1+\mathrm{deg}v)\ge |V(G){|}^2/(|V(G)|+2|E(G)|)$. Randomly order its vertices and retain each vertex preceding all its graph neighbours. This gives an independent set with the first expression as its expected value. The second inequality is the Cauchy-Schwarz inequality.

Carrier-sense multiple access lets a station attempt transmission only when interfering neighbors are inactive. An ideal continuous-time model gives inactive station $r$ an exponential attempt rate $e^{{\theta }_r}$ and unit-rate exponential transmission completions. Detailed balance then gives activation weights $e^{\theta \cdot n}$ on the independent sets of the interference graph.

For the binomial random graph $G(n,1/2)$, the chromatic number is $(1+o(1))n/(2{\log }_{2}n)$ with high probability. A first moment method bounds the independence number above. Edge-disjoint clique packing in the complement graph, followed by an edge-exposure martingale and a union bound over moderately large vertex sets, supplies independent sets for greedy colouring by removing independent sets.

The clique conflict graph has one vertex for each $r$-clique of a fixed graph, and joins two when they share an edge. Its independent sets correspond exactly to edge-disjoint clique packings. The Caro-Wei bound converts clique-count and overlap estimates into a packing lower bound.

A blow-up replaces each vertex of a graph by an independent set, and each original edge by all edges between the corresponding two sets. Replacing every vertex of $K_r$ by $t$ vertices gives the balanced complete multipartite blow-up $K_r(t)$.

The graphical Stirling number ${\left\{\begin{array}{c}n\\ k\end{array}\right\}}_G$ counts partitions of the vertices of a graph $G$ into $k$ nonempty independent sets. Equivalently, it counts proper colourings with $k$ unlabeled nonempty colour classes.

If every vertex subset of size at least $m$ contains an independent set of size $r$, repeatedly colour and remove an $r$-element independent set while at least $m$ vertices remain. Giving each remaining vertex its own colour uses at most $|V|/r+m$ colours.

Every finite graph of maximum degree at most $\Delta$ has an independent set of size at least $|V|/(\Delta +1)$. Greedily choose a vertex and delete its closed neighbourhood.

A subset $I$ of the vertices of a hypergraph is independent when no hyperedge lies wholly inside $I$. This generalizes an independent set in an ordinary graph. If a hypergraph has ordinary graph edges as its vertices and forbidden copies as its hyperedges, these independent sets encode graphs without those forbidden copies.

The independence number is the largest cardinality of an independent set in a graph. Since every colour class is independent,

An interference graph has one graph vertex for each station and an edge for every pair that cannot transmit simultaneously. Feasible activation schedules are the graph's independent sets.

A graph is $r$-partite when its vertices can be partitioned into $r$ classes, each an independent set. Edges may be missing between different classes. If all possible edges between different classes occur, it is a complete multipartite graph.