Around the Complete Intersection Theorem

In their celebrated paper, Erdos et al. (1961) posed the following question. Let F be a family of k-element subsets of an n-element set satisfying the condition that |F∩G|≥ℓ holds for any two members of F where ℓ≤k are fixed positive integers. What is the maximum size |F| of such a family? They gave a complete solution for the case ℓ=1: the largest family is the one consisting of all k-element subsets containing a fixed element of the underlying set. (One has to suppose 2k≤n, otherwise the problem is trivial.) They also proved that the best construction for arbitrary ℓ is the family consisting of all k- element subsets containing a fixed ℓ-element subset, but only for large n's. They also gave an example showing that this statement is not true for small n's. Later Frankl gave a construction for the general case that he believed to be the best. Frankl, Wilson and Füredi made serious progress towards the proof of this conjecture, but the complete solution was not achieved until 1996 when the surprising news came: Rudolf Ahlswede and Levon Khachatrian have found the proof. They invented the expressive name: Complete Intersection Theorem. We will show some of the consequences of this deep and important theorem. © 2016 Elsevier B.V

Complex surface singularities with integral homology sphere links

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities of "splice type", generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link Sigma should be a 4-manifold canonically associated to Sigma. We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We also point out recent counterexamples to some overly optimistic earlier conjectures in [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann. 326(2003) 75--93].Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper18.abs.htm