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Concept

Extremal problem

A question asking how large or small a structure can be while keeping some property. The answer usually has two halves that are proved separately: a bound saying nothing can beat some number, and a construction reaching it.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Everybody's share of the 24 chains. The subsets of a set of 4, each labelled with the fraction of maximal chains it lies on; the shares of any antichain add to at most one, and to exactly one only for a whole layer.

Everybody's share of the chains

There are twenty-four ways to build a four-element set one element at a time. Every subset lies on some of them, and no two incomparable subsets share one — so an antichain is a set of disjoint shares of a single whole.

discrete · Posets At most 3 of the 7 arcs can pairwise meet. The 7 elements arranged round a circle with the 7 arcs of 3 consecutive drawn, and the largest collection of arcs that pairwise intersect picked out.

The largest family that always meets

Change the question from "no two comparable" to "every two share an element" and the answer changes shape. The best antichain is a whole layer; the best intersecting family is a star, and the proof is a circle.

discrete · Posets Area at equal width: the triangle least, the circle most. A bar for each curve of constant width the family draws, all at the same width, with the bar's length its enclosed area and the extremes marked.

The least area a width can hold

Barbier's theorem says every curve of constant width has the same perimeter, which removes perimeter as a way of telling the family apart. Area is not like that — the circle holds the most and the Reuleaux triangle the least — and the reason the minimiser has corners is a constraint rather than a preference.

geometry · Constant width The largest hexagon of diameter one. Regular hexagon of diameter 1: area 0.649519053; best hexagon: area 0.674981443, 6 pairs at distance 1; vertices (0.3438, -0.3253) (0.5000, 0.2115) (-0.0000, 0.6138) (-0.5000, 0.2115) (-0.3438, -0.3253) (-0.0000, -0.3862).

The largest polygon of diameter one

Among pentagons whose points are never more than one unit apart, the regular pentagon has the most area, and the same is true for every odd number of sides. For six sides it is false: a lopsided hexagon found in 1975 beats the regular one by four per cent. Symmetry wins half the cases and loses the other half, and the reason is parity.

geometry · Isoperimetric

Named alongside it

The objects these essays reach for when they reach for this one.

Binomial coefficientConstant widthCounting argumentCounting two waysPosetAntichainAreaChainConvexityCyclic orderDiameterErdos ko rado

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