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Mathematics professor at Cambridge

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18
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Apr
30
asked Finding lots of discrete vectors in fairly general position
Apr
25
awarded  Good Answer
Apr
3
revised Euler-Lagrange equation for several occurrences of integrals
Corrected spelling in title
Apr
3
comment Almost orthogonal vectors
Looking at this almost a further year later, I'm still confused by Bill's remark, because what I wrote in the previous comment seems (i) correct and (ii) the standard volume argument that he discusses. Can anyone shed light on this?
Mar
31
comment What are the Applications of Hypergraphs
Here are two partial explanations for why algorithms based on hypergraphs are less common than algorithms based on graphs. 1. Some polynomial-time algorithms for graphs turn into NP-complete problems when you try to generalize them to hypergraphs (e.g., finding a perfect matching). 2. We often use graphs to model symmetric binary relations, and symmetric binary relations appear much more frequently than symmetric ternary relations (and beyond).
Mar
31
comment functions satisfying “one-one iff onto”
Does $f$ have to be continuous, or something like that? Otherwise, the result seems to be trivially false because you can mess about with the map on a set of measure zero.
Mar
30
comment Family of subsets such that there are at most two sets containing two given elements
It is perhaps worth adding that the above construction is generated by two standard tricks. The first is to dualize the problem by defining $T_i$ to be the set of $k$ such that $i\in S_k$ and reformulating the conditions in terms of the $T_i$. (The main one says that the maximum intersection of any two $T_i$ is 2.) The other trick is to use graphs of polynomials to get plenty of sets with small intersections.
Mar
30
comment Family of subsets such that there are at most two sets containing two given elements
Thanks a lot -- I've edited it now.
Mar
30
revised Family of subsets such that there are at most two sets containing two given elements
edited body
Mar
30
answered Family of subsets such that there are at most two sets containing two given elements
Mar
27
awarded  Good Answer
Mar
25
comment Family of subsets such that there are at most two sets containing two given elements
Have you tried taking the characteristic functions of the $S_i$, adding them up, and looking at the $\ell_2$ norm? The condition on the $S_i$ should put a strong condition on the average inner product, and then the Cauchy-Schwarz inequality should give a bound the other way. I feel this ought to work, but can't quite be certain without writing it down.