Extremal graph
Named by 5 essays across one field — each of them below, with the objects they name alongside it.
The edge that forces a triangle
A graph on six points can carry nine edges with no three of them closing a triangle. It cannot carry ten. The bound is n²/4, the graphs that achieve it are all the same shape, and both facts fall out of examining every graph there is.
Moves that only ever add edges
Turán's theorem says the densest graph avoiding a complete graph on r + 1 points is the balanced r-part graph. Zykov's proof finds it by a sequence of moves — turn a point into a copy of a better-connected one it is not joined to — each of which adds edges and none of which can create the forbidden clique. A second proof spreads a unit of weight over the points and gets the same bound from a maximum.
The densest graph without a square
Forbid four points joined in a cycle and a graph can keep only about ½n^(3/2) of its edges — far fewer than the quarter of all pairs a triangle-free graph keeps. Counting pairs of neighbours proves the ceiling in two lines. What reaches it is not a random graph but a finite geometry: the points of a projective plane, joined when they are orthogonal.
The fewest triangles an edge density allows
Half of all possible edges can be drawn without a single triangle. One more, and triangles appear — not one but several at once. Push the density further and the question becomes a curve: for every share of edges, the fewest triangles a large graph can hold. The answer is a string of scallops, touching a simple parabola at the densities of the balanced multipartite graphs and bulging above it between them. It was guessed in the 1980s and proved in 2008 by a method that turns counting into positive-definite matrices.
As many points as two steps allow
In a graph where every point has d neighbours and every point is within two steps of every other, there can be at most d² + 1 points — one, its d neighbours, and d(d − 1) more reached through them. Graphs that meet the bound exactly are rare to the point of absurdity. The pentagon does it for d = 2, the Petersen graph for d = 3, a fifty-point graph found in 1960 for d = 7, and an eigenvalue argument proves there is nothing else — except possibly one graph with 3,250 points and 57 neighbours each, which nobody has found or ruled out.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchComplete bipartiteEdge countTuran graphCliqueConvexityProjective planeTriangle-freeCounting two waysEdge densityEigenvalueFinite field