One striking example of the failure of Hamilton-Ivey pinching can be seen here in which it is shown that the FIK shrinkers (which do not have non-negative Ricci curvature, much less sectional curvature, can arise as blowup limits to the Ricci flow).

As far as I know, basically all that is known is that ancient solutions to the Ricci flow (e.g. blowup limits) have non-negative scalar curvature, by Corollary 2.5 here

One striking example of the failure of Hamilton-Ivey pinching can be seen here in which it is shown that the FIK shrinkers (which do not have non-negative Ricci curvature, much less non-negative sectional curvature), can arise as blowup limits to the Ricci flow.

As far as I know, basically all that is known is that ancient solutions to the Ricci flow (e.g. blowup limits) have non-negative scalar curvature, by Corollary 2.5 here

One striking example of the failure of Hamilton-Ivey pinching can be seen here in which it is shown that the FIK shrinkers (which do not have non-negative Ricci curvature, much less sectional curvature, can arise as blowup limits to the Ricci flow.

As far as I know, basically all that is known is that ancient solutions to the Ricci flow (e.g. blowup limits) have non-negative scalar curvature, but Corollary 2.5

One striking example of the failure of Hamilton-Ivey pinching can be seen here in which it is shown that the FIK shrinkers (which do not have non-negative Ricci curvature, much less sectional curvature, can arise as blowup limits to the Ricci flow).

As far as I know, basically all that is known is that ancient solutions to the Ricci flow (e.g. blowup limits) have non-negative scalar curvature, by Corollary 2.5 here