Szemerédi's theorem is a fundamental result in combinatorial number theory, established by mathematician Endre Szemerédi in 1975. It states that any subset of the integers with positive density contains arbitrarily long arithmetic progressions. This means that if you have a large enough collection of numbers, you can always find sequences where the numbers are evenly spaced apart. The theorem has significant implications in various areas of mathematics, including graph theory and ergodic theory. It extends to more complex structures, such as higher-dimensional spaces, and has inspired further research into patterns within sets of numbers, influencing the study of additive combinatorics.