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Small Numbers

This repository is a shrine to small numbers.

Specifically, we are interested in concrete values of combinatorial functions with small parameters, especially those in the world of extremal combinatorics and Ramsey theory.

Many people focus on the asymptotic behavior of these functions when the parameters are arbitrarily large. Instead, we focus on the details of small cases. Interesting things happen in these small values.

Here are a few such functions:

  • Szemerédi Numbers: The number sz(k,n) is the size of the largest subset A of {1,...,n} that does not contain an arithmetic progression of length k.

  • van der Waerden Numbers: The number vdw(t,k) is the smallest n such that every t-coloring of {1,...,n} contains a monochromatic arithmetic progression of length k. There are off-diagonal versions of these numbers, too.

  • Erdős Discrepancy Numbers: The number disc(C) is the smallest n such that every sequence {x_1,...,x_n} with each x_i in {-1,+1} has discrepancy at least C. That is, there exist integers d and k with d*k <= n where the absolute value of the sum x_d + x_2d + ... + x_kd is at least C. Only disc(2) and disc(3) are known.

Approach

We collect results from the mathematical literature to create a list of the known values and bounds for these small numbers.

We try to reproduce these results using "simple" approaches.

One such simple approach is to use a SAT solver. In our case, we will use the Z3 Satisfiability Modulo Theory (SMT) solver. There are some very powerful proof techniques built into this tool, and we can use them to find bounds and prove their value. We provide these proofs for later verification when possible.

Another simple approach is to use simple backtrack search with some basic constraint solving methods. These can sometimes provide easier to understand approaches to these calculations, but also to enumerate all extremal solutions.

License

Small Numbers is available as open-source under the MIT License. See LICENSE.md for more details.

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A collection of combinatorial functions and their values on small numbers

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