Bertrand’s postulate

In 1845 Joseph Bertrand (1822–1900) conjectured that there is always at least one prime between n and 2n for n ≥ 2. Bertrand himself verified the statement for all numbers in the interval 2 < n < 3,000,000. The conjecture was proved by Pafnuty Chebyshev (1821–1894) in 1852. A simpler proof using the properties of the Gamma function was later provided by Ramanujan in 1919.

At age 19, Erdős in 1932 published his first paper, providing a surprising elementary proof using binomial coefficients and the Chebyshev function ϑ(x). The paper, entitled Beweis eines Satzes von Tschebyschef (“On a proof of a theorem of Chebyshev”) was published in Acta Scientifica Mathematica. His proof considers the middle binomial coefficient:

Equation 1. The binomial coefficient

A lower bound is:

Equation 2. A lower bound for the binomial coefficient in equation 1

Indeed, the binomial coefficient in equation 1 is the largest term in the 2n+1-term sum:

Equation 3.

The first part of Erdős’ proof shows that if there is no prime p with n < p ≤ 2n, then we can put an upper bound on the binomial coefficient that is smaller than 4ⁿ / (2n +1) unless n is “small”. This verifies Bertrand’s postulate for all sufficiently large n. The second part deals with small the cases where n is “small”. These are dealt with by hand. For a narration of these cases, see Galvin (2015).

The Prime Number Theorem

In July 1948, Erdős met Norwegian mathematician Atle Selberg at the Institute for Advanced Study. From their brief encounter, an elementary proof of the Prime Number Theorem appeared (Babai, 1996). Originally emerging as a consequence of the independent works of Legendre, Gauss and Dirichlet, the prime number theorem states:

Equation 3. The Prime Number theorem states that as x goes to infinity, the prime counting function π(x) will approximate the function x/ln(x).

The theorem was famously independently proved by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using the Riemann zeta function. Selberg in March 1948 established that the asymptotic formula

Equation 4. Selberg’s formula

Where the Jacobi theta function ϑ(x) is equal to the sum of the log of primes less than or equal to x and O(x) is an upper bound for x expressed in Big O notation. By July, both Selberg himself and Erdős had used Selberg’s formula to prove the prime number theorem. Who of the two proved the result first became somewhat of a priority dispute (Goldfeld, 2004), leading the two to unfortunately never collaborate again.

Ramsey theory

Of Erdős’ most consequential results, his contributions to the development of Ramsey theory clearly stand out. Ramsey theory is the branch of mathematics concerned with studying the ‘conditions under which order must appear’. A typical example of a such a problem starts out with a mathematical structure (such as a graph), which is then cut into pieces. A typical question is “How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property?”

The Erdős-Szekeres Theorem (1935)

The Erdős-Szekers theorem makes precise one of the corollaries of Ramsey theory, namely that given r and s any sequence of distinct real numbers with length at least (r - 1)(s - 1) + 1 contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s. The result was first shown in Erdős and Szekers’ influential 1935 paper A combinatorial problem in geometry:

The Erdős-Szekeres theorem (1935)

Any real sequence of at least ad + 1 terms contains either an ascending subsequence of a + 1 terms or a descending subsequence of d + 1 terms.

A subsequence is a sequence that can be derived from another sequence by deleting some of the elements without changing the order of the sequence. For instance, given the sequence ABCD, its subsequences are ABC, BCD, AB, BC, CD, AC, AD, BC and BD. Given r = 2 and s = 2:

Example:

For r = 2 and s = 2, the formula tells us that any permutation of three numbers has an increasing subsequence of length three or a decreasing subsequence of length two. Among the six permutations of the numbers 1,2,3: • 1,2,3 has an increasing subsequence of all three numbers

• 1,3,2 has a decreasing subsequence: 3,2

• 2,1,3 has a decreasing subsequence: 2,1

• 2,3,1 has two decreasing subsequences: 2,1 and 3,1

• 3,1,2 has two decreasing subsequences: 3,1 and 3,2

• 3,2,1 has three decreasing subsequences: 3,2, 3,1, and 2,1.

In their paper Erdős and Szekeres used proof by induction to show that f(n) = (n - 1)² + 1, where f(n) denotes the least integer such that any subsequence of f(n) real numbers must contain a monotone subsequence of length n. Steele (1995) later reviewed six different proofs of the same theorem, including the original proof by Erdős and Szekeres (1935), those by the pigeon-hole principle (Hammersley, 1972; Black, 1971; Seidenberg, 1959), one by one-to-one correspondence (Standon and White, 1986) as well as one that follows from Dilworth’s theorem (1950). The most widely cited proof is likely that of Hammersley (1972). The key idea of this proof by pigeon-hole is to place the elements of the sequence x₁, x₂, …, xₘ with m = n² + 1 into a set of ordered columns by the following rules: a) let x₁ start the first column, and b) for i ≥ 1, if xᵢ is greater than or equal to the value that is on top of a column, put xᵢ on top of the first such column, and c) otherwise start with a new column xᵢ (Steele, 1995).

There are two points of notice in Hammersley’s proof. The first is that the elements of any column correspond to an increasing subsequence. The second is that the only time we shift to a later column is when we have an element that is smaller than one of its predecessors. Thus, given k columns in the final construction, one can trace back from the last and find a monotone subsequence of length k. Since n² + 1 numbers are placed into the column structure, one must either have more than n columns or some column of greater height than n. Either way, there must be a monotone subsequence of length n + 1 (Steele, 1995).

The “slickest and most systematic” of the proofs, according to Steele, is that which “is naturally suggested by dynamic programming”, presented in a single page by Seidenberg (1959):

Proof of the Erdős-Szekeres theorem (Seidenberg, 1959)

Given a sequence of length (r - 1)(s - 1) + 1, label each number nᵢ in the sequence with a pair (aᵢ, bᵢ) where: • aᵢ is the length of the longest monotonically increasing subsequence ending with nᵢ and

• bᵢ is the length of the monotonically decreasing subsequence ending with nᵢ. Each two numbers in the sequence are labeled with a different pair: if i < j and nᵢ ≥ nⱼ then aᵢ < aⱼ and on the other hand if nᵢ ≥ nⱼ then bᵢ < bⱼ. But, there are only (r - 1)(s - 1) possible labels if aᵢ is at most r - 1 and bᵢ is at most s - 1. So, by the pigeonhole principle, there must exist a value of i for which aᵢ or bᵢ is outside this range. If aᵢ is out of range then nᵢ is part of an increasing sequence of length at least rᵢ. If bᵢ is out of range then nᵢ is part of a decreasing sequence of length at least s.

The Happy Ending Theorem (1935)

It was Erdős and Szekeres’ mutual friend Esther Klein who in 1932 first observed that:

The Happy Ending theorem

Any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral.

Points in general position in the plane are points which no three belong to a line. Convex quadrilateral are polygons with four edges and four corners whose interior angles are less than 180°, such as rectangles, rhomboids, trapezoids and parallelograms. The three distinct types of placements of five points in general position in the plane are (Morris & Soltan, 2016):

Any five points in general position in the plane determine a polygon with four edges and four corners whose angles are less than 180°

The problem statement and theorem was one of the first influential results that eventually led to the development of Ramsey theory. Erdős called the result the “Happy Ending Problem” because it eventually led to the marriage of Szekeres and Klein. The theorem is a particular case of the more general theorem proved by Erdős and George Szekeres in the same 1935 paper that proved the Erdős-Szekeres theorem of monotonically increasing and decreasing infinite subsequences, namely that:

Erdős & Szekeres' Generalization of the Happy Ending Theorem (1935)

For any positive integer n, any sufficiently large finite set of points in the plane in general position has a subset of n points that form the vertices of a convex polygon.

The Erdos-Szekeres conjecture (1935)

While the Erdős-Szekeres theorem (1935) proves the existence of the finite number g(n), in the same paper Erdős and Szekeres also conjectured what the number g(n) is:

The Erdős-Szekeres conjecture

The smallest number of points m for which any general position arrangement contains a convex subset of n points is 2ⁿ⁻² + 1.

The conjecture is known to hold for its known values of g(3), g(4), g(5), g(6). It is trivial to observe that g(3) = 3, i.e. that any three points in the plane that do not belong to a line form a triangle with interior angles less than 180°. Klein’s observation in the happy ending theorem is that g(4) = 5. That g(5) = 9 was first proven by Endre Makai, but first appeared in print in a proof by Kalbfleisch, Kalbfleisch and Stanton (1971). A computer-aided proof that g(6) = 17 was proved by Szekeres & Peters (2006). They carried out a computer search which eliminated all possible configurations of 17 points without convex hexagons. The value of g(n) is unknown for values larger than n = 6, and the Erdős-Szekeres conjecture still remains open.