Concept

Extremal configuration

An arrangement achieving the largest or smallest possible value of some count, given a constraint. Finding one settles half of an extremal problem; the other half is proving that nothing does better, and the two halves are usually different arguments.

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

The nine-point grid, and the lines they force. 9 points with all 20 of their connecting lines drawn. The 12 carrying exactly two points are drawn solid and the rest faintly; the count is computed from the coordinates rather than read off the drawing.

The line with only two points on it

Scatter finitely many points on a page, not all in one line, and draw every line through two or more of them. However cunningly the points are placed, some line ends up carrying exactly two — and the proof is a minimisation with no algebra in it at all.

geometry · ordinary lines

A sequence of 3² with no climb and no fall longer than 3. 10 terms plotted in order, each labelled with the longest climb and the longest fall ending at it; the first 9 keep both counters at 3 or below and the last one cannot.

The sequence that cannot avoid a staircase

Any ten numbers in a row contain four that climb or four that fall. The proof gives every term a pair of counters, notices that no two terms can share a pair, and is finished — with a bound that is exactly right.

discrete · ramsey theory

Named alongside it

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

Collinear Configuration Convex position Counting argument Erdos szekeres Existence proof Incidence Minimal counterexample Monotone subsequence Ordinary line Partial order Permutation

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