# Difference between revisions of "Find a good configuration of HAPs"

The rough idea is this. Recall that a hypergraph is a collection of subsets of some set X. We borrow graph terminology and call the elements of X verticesand the sets in H edges (or sometimes hyperedges if we want to make it clear that our graph is hyper). If H is a hypergraph with vertex set X and f is a function from X to {-1,1}, then the discrepancy of f is the maximum of $|\sum_{x\in A}f(x)|$ over all edges A of H. The discrepancy of H is the minimum discrepancy of any function $f:X\to\{-1,1\}$.
The Erdős discrepancy problem is asking us to prove that the discrepancy of the hypergraph whose vertex set is $\mathbb{N}$ and whose edges are all finite HAPs is infinite. One obvious strategy for doing this is to identify within this hypergraph a collection of hypergraphs $H_n$ that have properties that allow us to prove arbitrarily good lower bounds for the discrepancy. What makes this a strategy, rather than a trivial reformulation of the problem (after all, one might suggest that there is no point in taking all HAPs) is that one would be aiming to find subhypergraphs of the HAP hypergraph that lent themselves particularly well to certain kinds of combinatorial arguments.