## model theory – Roelcke precompactness and Ramsey property

A survey by Nguyen Van Thé (2014) has Conjecture 1,
which is that
“every closed oligomorphic
subgroup of $$S_∞$$ should have a metrizable universal minimal flow with a generic
orbit.” Later, it goes on to say that “it is even possible that this should be
true for a larger class of groups, called Roelcke precompact.” (Let me call this Conjecture 1′.) Now, Kwiatkowska (2018) exhibited a group without a metrizable universal minimal flow that is not Roelcke precompact, so we need to stick to Conjecture 1.

How about the converse of Conjecture 1′, i.e., the statement that a closed subgroup of $$S_∞$$ is Roelcke precompact if it has a metrizable universal minimal flow? Is there a proof or a counterexample? In absence of either of the two, do people believe it?

## ramsey theory – Equal subset-sums of bounded vectors

Let $$Ssubseteq {0,ldots,n}^d$$ be a set of $$d$$-dimensional vectors of with bounded, natural, coordinates.

We are given that
$$v’+v_1+ldots+v_t=u’+u_1+ldots+u_s$$
where $$v_1,ldots,v_t,u_1,ldots,u_s,v’,u’in S$$ (and the vectors are not necessarily distinct).

That is, two sets of vectors whose sums are equal.

I want to prove that, if $$t$$ and $$s$$ are large enough, then there
exist subsets $$Isubseteq {1,ldots,t}$$ and $$Jsubseteq {1,ldots > s}$$ such that $$sum_{iin I}v_i=sum_{jin J}u_j$$

Note that, without the assumption of the equal sum above, there may not be such sets (e.g., if all the $$v_i$$ are $$(1,0)$$, and all the $$u_i$$ are $$(0,1)$$). Also, for small $$s,t$$ there may not be such sets.

Some informal thoughts:
My intuition is that for large enough $$s,t$$, we can force a lot of repetitions of vectors within the sets, and then we can “tailor” equal sums. This is somewhat akin to a vector version of Erdős-Ginzburg-Ziv (or the Van Emde Boas – Kruyswijk variation, which looks at vectors), but instead of looking at the finite abelian group, I have the sum above to bound the behaviour.

Also, I don’t really care about tight bounds for $$s,t$$. They can be as large as needed (e.g., exponential, or even double exponential in $$|S|,n$$ is fine).

## co.combinatorics – Reference – Lower limit of probability threshold for Ramsey properties

I look at a paper by Rodl and Rucimski: Threshold function for Ramsey properties, also available semantically scholarly. There they indicate the following:

For all integers $$r geq 2$$and for each chart $$G$$ What is not a star forest is constant $$c$$ and $$C$$ so that
$$lim_ {n to infty} mathbb {P} (K (n, p) to (G) _r ^ 2) = left { begin {array} {ll} 0 & text {if} pCn ^ {- 1 / m_G ^ {(2)}} end {array} right.$$

Where $$K (n, p)$$ is the usual binomial random graph model, and $$H to (G) _r ^ 2$$ the usual Ramsey notation, which means that none $$r$$– coloring the edges of $$H$$ contains a monochromatic $$G$$,

They prove the 1 statement in the same paper. The 0 instruction is written

The proof of the 0 statement from Theorem 1 & # 39; appeared in (RR93)

And the reference is

(RR93) _, Lower bounds for probability thresholds for Ramsey properties, Combinatorics, Paul Erdos is eighty (Volume 1) Keszthely (Hungary), Bolyai Soc. Mathematics. Studies, 1993, pp. 317-346

I found only one source for this paper, Rucinski's own website.

There, however, the 0 statement is not proven for any number of colors, only for 2 (or I missed the generalization). The proved sentence is

For every diagram $$G$$ What is not a star forest, there is a positive constant $$c_G$$ so that
$$lim_ {n to infty} mathbb {P} (K (n, cn ^ {- 1 / m_G ^ {(2)}}) to (G) _2 ^ 2) = 0$$

The paper contains only one final note

Added as proof. Recently the authors have proven sentence 3 for any number of colors.

I found a dozen papers related to that full result (I.e., $$r$$ Colors), all with the same references, but not the actual paper.

Can someone point me to a place where I could find him?

To edit: I should add that I was unable to access the originally quoted magazine "Paul Erdos is Eighty Vol1" online or in print.

## Ramsey Number \$ 40 leq R (3,10) leq \$ 42

I read about Ramsey's numbers and the tests that have been done so far. I know that it is almost impossible to know the exact value of brute force $$R (3.10)$$ that is limited by $$40$$ and $$42$$,