I am interested in using Shannon’s entropy in combinatorics. It is often presented with a motivation of how much information can be passed, but assume I am not interested in that, I want to understand why it is useful as a tool to bound objects (with examples now).
Several examples that demonstrate that very nontrivial results follow from relatively entropy arguments are:
- Bregman’s inequality about permanents.
- Whiteny’s inequality about volume and its projections.
- Shearer’s lemma and numerous consequences; bounds on the number of independent sets in bipartite d-regular graph, number of homomorphisms to graph…
- The largest size of an intersection unique family (see https://link.springer.com/article/10.1007/BF02579460 ).
If you could help me with the following I’d be very happy;
How would someone just interested in counting and bounding combinatorics arrive at entropy?
A common theme (noted by my proffessor when teaching about it) to some of the results are that original proofs used convexity arguments, how healthy easy it to be think of entropy in combinatorics as a clean way to state convexity arguments and deriving nontrivial results by assigning weights to objects and bounding sums?
When can proofs be translated to entropy proofs? For instance I find it strange there is the $ k-intersection$ proof with entropy, while I’m not aware of an entropy proof for sperner’s theorem about antichains.
When are problems suspectible to entropy methods?
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