Concept

Convex position

A set of points is in convex position when every one of them is a corner of their convex hull, so that none lies inside the polygon the others form. Asking how many points force a subset in convex position is the Erdős–Szekeres problem.

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

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

12 points, a cup of 5 and a cap of 5. 12 points in general position with the longest convex-upward chain (5 points) and the longest convex-downward chain (5 points) marked.

Every crowd holds a bowl or a dome

Among enough points in the plane, some k of them always bend upward like a bowl or some l bend downward like a dome. The number that forces it is a binomial coefficient, it is exactly right, and it proves that every large enough crowd contains a convex polygon — with a bound nobody could lower to the true answer for eighty years.

discrete · Ramsey theory

Named alongside it

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

Pigeonhole principle Binomial coefficient Counting argument Erdos szekeres Exhaustive search Existence proof Extremal combinatorics Extremal configuration General position Monotone subsequence Partial order Pascals triangle

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