# Progress on Borsuk's Problem
Borsuk's Problem is a classic question in combinatorial geometry that asks how many pieces a shape must be divided into so that each piece has a smaller diameter than the original. The key function here is f(d), which represents the smallest number of sets needed to cover any diameter-one shape in d-dimensional space, where each covering set has a strictly smaller diameter. A related function, g(d), captures a similar idea with slight variations, and understanding the relationship between these two functions is central to the research discussed in this post.
For decades, mathematicians believed that f(d) grew polynomially with dimension, but a landmark result in the 1990s showed that the answer actually grows exponentially, meaning shapes in high dimensions are far harder to break apart than anyone expected. The problem remains active today, with researchers working to pin down the precise growth rate of f(d) and g(d) and to determine whether the finite and infinite versions of the problem behave differently from one another.
This post also addresses the answer to TYI 60, a reader challenge connected to these functions, inviting the math community to engage directly with open problems at the frontier of discrete geometry. The interplay between finite coverings and infinite geometric structures makes Borsuk's Problem a rich and rewarding area of study, combining elegant intuition with surprisingly deep complexity. Progress in this field continues to reveal just how much geometric structure remains hidden even in familiar spaces.