In 1973, Paul Erdős asked if it was possible to assemble sets of “triples”—three points on a graph—so that they abide by two seemingly incompatible rules.
The goal here is to trace out triangles on top of these lines such that the triangles satisfy two requirements: First, no two triangles share an edge. And second, ensure that every small subset of triangles utilizes a sufficiently large number of nodes.Say that instead of making triangles out of edges, you’re building houses out of Lego bricks. The first few buildings you make are extravagant, with structural reinforcements and elaborate ornamentation.
In the case of the Steiner triple system, you’re trying to create triangles. Your absorber, in this case, is a carefully chosen collection of edges. If you find yourself unable to sort the rest of the system into triangles, you can use some of the edges that lead into the absorber. Then, when you’re done doing that, you break down the absorber itself into triangles.
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