In set theory there are many types of cardinal numbers. One of them is the weakly compact cardinal. Wikipedia states the following about it:
Formally, a cardinal $\kappa$ is defined to be weakly compact if it is uncountable and for every function $f: [\kappa]^2 \rightarrow \{0, 1\}$ there is a set of cardinality $\kappa$ that is homogeneous for $f$. In this context,
$[\kappa]^2$ means the set of $2$-element subsets of $\kappa$, and a subset $S$ of $\kappa$ is homogeneous for $f$ if and only if either all of $[S]^2$ maps to $0$ or all of it maps to $1$.
What exactly does this mean? Is there a simpler way to define a weakly compact cardinal?
One intutive-'ish' way to think of this definition is in terms of coloring a complete graph of size $\kappa$. The function $f$ represents a coloring of the edges of this graph (i.e., every size-$2$ subset of $\kappa$) with two colors, and the set $S$ then represents a $\kappa$-sized subgraph of the complete graph such that all of the edges between elements of $S$ have the same color. This starts to break down for more complicated partition principles, but for one like this it's the very rough mental model that I use.