The Six Degrees of VDW

 A long long time ago  a HS student, Justin Kruskal (Clyde's  son)  was working with me on upper bounds on some Poly VDW numbers (see here for a statement of PVDW). His school required that he have an application.  Here is what he ended up doing: rather than argue that PVDW had an application he argued that Ramsey Theory itself had applications, and since this was part of Ramsey Theory it had an application. How many degrees of separation were there from his work and the so called application.

1. The best (at the time) Matrix Multiplication algorithm used 3-free sets.
2. 3-free sets are used to get lower bounds on VDW numbers.
3. Lower bounds on VDW numbers are related to upper bounds on VDW numbers
4. Upper bounds on VDW are related to upper bounds on PVDW numbers.

Only 4 degrees!  The people in charge of the HS projects recognized that it was good work and hence gave him a pass on the lack of real applications. Or they didn't quite notice the lack of applications. He DID end up being one of five students who got to give a talk on his project to the entire school. When you say that your work has applications is it direct? one degree off? two? Are all theorems no more than six degrees away from  an application? Depends on how you define degree and application.