TITLE: Hamilton-Ivey pinching in dimension 4
QUESTION [6 upvotes]: I've heard it said (e.g., in the accepted answer to this MO question) that a major obstacle to an effective theory of Ricci Flow in dimension 4 is the absence of the Hamilton-Ivey pinching phenomenon. I'm curious about the possibilities for such a pinching in dimension 4, but I couldn't locate any information about it. I'm curious about 2 complementary questions in this regard.

Are there any known partial results or indications of what such a pinching may look like in dimension 4?
Are there known examples that constrain the form of or throw doubt upon such a possible pinching?

Any thoughts or references to the literature are welcome.

REPLY [2 votes]: http://arxiv.org/pdf/0807.1582.pdf
In this paper, the author gave a local result of Hamilton-Ivey pinching on the gradient shrinking soliton ($n\ge 4$) with vanishing Weyl tensor (see prop 3.2). Also in the introduction part, the author announced a pinching result for the solution to Ricci flow in $n-$dimension provided the solution is LCF for all time, although I believe the proof has not been published yet.