[skinner_]

Sorry, I've just spotted your question 3 days late. The Moser-Erdős question is about how large the density of a unit distance avoiding set can be. The Hadwiger-Nelson question is about how many unit distance avoiding sets are necessary to cover the whole plane.

As David Eppstein phrased it (https://mathstodon.xyz/@11011110/110810118775049296), ours is the independent set version of Hadwiger-Nelson. That is, the relationship between the two questions is analogous to the relationship between graph chromatic number and graph independence number.

If you define independence ratio as the ratio of the independence number and the vertex number, then this is a bit more than just an analogy. If you build a huge grid of very tiny squares, and connect two tiny squares with an edge if there's an unit distance between them, then in the limit, the independence ratio of this graph is the density that we've looked at, and the chromatic number of this graph is the measurable chromatic number of the plane.

[dabeddabed]

Don't worry I spotted your answer 4 days late. Had to read the intro to the paper to understand what you were pointing to me (in fact maybe I should have tried before asking my first question) only section 1 the rest is beyond me, if I understood right we have that,

m_1(R^2)<=1/X_f(R^2)<=1/X_f(G)<=a(G)/|G|.

What I'm not sure of is how X_f(R^2) relates to X(R^2) that is if we were to find that the chromatic number of the plane is X(R^2)=6 then what can we say about fractional chromatic number X_f(R^2)?

Hope you see this post and if not it was still super useful, thanks a lot!

[skinner_]

The fractional chromatic number of the plane is by definition less than or equal to the regular chromatic number of the plane, but already we know for sure that they are not equal. We know that 5 <= X(R^2) <= 7, and we know that 3.8991 <= X_f(R^2) <= 4.3599 [1]. Jaan Parts has unpublished results indicating that 3.9898059 <= X_f(R^2). Our team believes that it is not a coincidence that the long list of X_f(R^2) lower bounds seems to converge to 4, and we conjecture that X_f(R^2)=4. This conjecture would have the interesting implication that our result on m_1(R^2) cannot be obtained via bounding 1/X_f(R^2), and the harmonic analysis apparatus that we use is fundamental to proving Erdős's conjecture.

[1] https://www.sfu.ca/~vjungic/RamseyNotes/Fractional.html