Homomorphism
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
The only bit that survives
A shuffle can be called even or odd, and the label behaves under composition. Ask whether some cleverer label — a number out of three, or out of four — could behave the same way, and the answer is that nothing else can — one bit is exactly what a permutation gives up.
No graph is poorer in squares than chance
Take any graph with a fixed share of its possible edges and count how often four points chosen in order close up into a ring. However the edges are arranged — in cliques, in two-sided blocks, at random — the count never falls below what a random graph of the same density gives. Sidorenko conjectured that the same holds for every pattern without an odd cycle. For a triangle it is false; for a path or a ring of four it takes two lines; for K₅,₅ with a ten-cycle removed nobody knows.
Named alongside it
The objects these essays reach for when they reach for this one.
Abelian groupBipartite graphCauchy schwarzCommutatorConjugacy classCounterexampleDensityExhaustive searchExtremal graph theoryInvariantNormal subgroupPermutation