Call for nominations for the 2018 Chern Medal | Gowers

]]>Maybe a conjecture could be of the form :

For any Sidon set , with , such that there exists , such that ,

Where

_ is a fonction that we have to find , that is close to identity

_ is a set of Sidon sets with some quadratic property that we have to precise

_ is a increasing fonction from to , where is order relations that we also have to precise (for example we can take iff )

_ is such that is small when is small.

The idea is that if we get close to we can find a Sidon set with a quadratic structure close to . But maybe it is not enough and you want to have itself the "quadratic property", not just close to a set that has it….

We can also state a "skeleton" of direct conjecture – that could maybe help to find/solve the inverse problem that we try to formulate – as :

For any , there exists such that

where is close to when is small (for )

As I'm not familiar with the problem, I won't take the risk to give explicit , , so I don't really "say something" neither. But just a direction, so that we might be able to discuss on the choice of each parameter.

Take if and if not, where has no f-intersection. ]]>

if there exists such that :

for all

(the decreasing of is quicker than that of , that's why the inequality is this way, that is also why I used the "greater or equal " instead of switch the places of A and B, if the bug doesn't come in this very comment, this will show that it does'nt react to "lt" the same way tan to "gt". *)

*by the way… is it " a > b" or " a \> b " that works ?

Well, I'm going to test it, so that you don't loose time by answering that kind of boring details :

test

(with the barre)

(without the barre)

There is kind of an “inverse problem” by asking who are the partial orders such that .

I say again the definition of :

for all integer, where is the usual order on and $A^*$ is the increasing injection from to $A=A^*(\mathbb{N})$

(the previous definition took $a\in \mathbb{Z}$ so there is some definition problem for the fonction if )

Let's also say again the two interesting fact that we have with

1) it is extending and compatible with (where )

2) and

So that 2) get rid of the problem we had before with that is not true… we pay this nice fact by dealing with instead of but there may be hope to get back to with new statements that are interesting independently from $p=t$.

One of this statement could be the fact that $p_{fct}=p$ that I spoke about at the beginning of the post but there is another one (that might be very wrong, but we need a weaker version)

monotone no-intersection statement :

Suppose that is a collection of families such that , for all and suppose that no $F^k$ have a non-empty f-intersection, then have no f-intersection as well.

This statement, if true, would help us in the particular case of and that $F=F_0$ has the property that any ending segment has no-intersection ( $\left\{F^0_i,\, j<i<p\right\}$ has no f-intersection for all )

There is a construction mixing the usual argument in the main comment, the "more and more tower-like" idea in the next main comment, and the $*$ argument that failed because of not being true :

If is successor we say if , and .if not

If is limit, we take any f-intersection of and we say that if , and if we take .

We can imagine other construction, but if the statement is true, and if we can suppose that we can assume the existence** of the particular case of a family with cardinality where the ending segment have no-intersection (I think this** is not difficult) then is a tower with no-intersection.

So if either monotone no-intersection statement (and **) is true either is true then we have what we want.

]]>I try to limit the number of posts but I think it is really necessary here, to write it again!)

Let’s say that elements of are semi-closed subset of ground-set , if is closed under the intersection and if it contains and ground-set . Let’s then define , the “semi-adherence” of any as the smallest semi-closed set that contain . If is closed under intersection, then for all .

For any $F\subset \mathcal A$, is the “adherence class” of and it’s a -up-set :

– meaning for all , we have and . Then, as an up-set , satisfied Frankl Union conjecture condition in a trivial way : (the fonction from i s an injection).

If this injection in a bijection, then let’s say that is -saturated. And note by (or just ) the set of -saturated couples. It is easy to see that or each , is a surjective map from to – where is the set of all such that

This shows that for any we have so that any family of semi closed subset of , satisfies “Frankl Intersection up to saturation” for EVERY .

This is quite a surprise for me because despite the fact that being “saturated” seems to be quite a strong constraint, it’s also easy to identify, and quite “clean” and simple , we can hope that it can help us doing some statement and calculus that might be relevant.

We can also hope for a generalization as follow :

First we observe that if and only if for any maximal for (the inclusion) in , we have . Then we can use this definition to be the definition of a saturated couple , in a general (not especially closed under intersection, of course If it is closed under intersection the two definition are equivalent)

With this general definition we can ask ourselfs if we can say something about the ratio

Is there a $\epsilon>0$ such that this ratio never get greater then for any that is closed under intersection? same question for a general family, with the general definition of …?

If the answer is “yes” to one of these questions then, the weak Frankl is true – meaning there exists such that for any closed under intersection, $ latex \inf_{x\in S}|\mathcal{A_x}|/|\mathcal{A}| $ is never greter then .

]]>Maybe it is well known already….but I never red it yet…

It takes few definitions inspired from topology, but it’s, then, quite nice and natural… I feel more confortable with the “intersection Frankl”, but we have obviously the corresponding statement for FUNC, so I use Intersection, to reduce the risk to make notations mistakes.

Let’s say that elements of semi-closed subset of ground-set $S$, if is closed under the intersection and contain and ground-set $S=latex \bigcup{\mathcal{A}}$ . Let’s then define , as the semi-adherence of any as the smallest semi-closed set that contain . If $\mathcal A$ is closed under intersection, then for all . For any $F\subset \mathcal A$, is the “adherence class” of and it’s a -up-set :

– meaning for all , we have and $F=\bigcup{\mathcal{A}}$. Then, as an up-set it satisfied Frankl Union conjecture condition in a trivial way : (the fonction from to is an injection).

If this injection in a bijection, then let’s say that is -saturated. And note by the set of $\mathcal{A}$-saturated couples. It is easy to see that or each , is a surjective map from to – where is the set of all $X$ such that .

This shows that for any $x\in S$ we have so that any family of semi closed subset of , satisfies “Frankl Intersection up to saturation” for EVERY .

This is quite a surprise for me because the fact to be “saturated” seems to be quite a strong constraint, and it’s also easy to identify and we can hope that it can help us doing some statement and calculus that might be relevant.

We can also hope for a generalization as follow :

First we observe that if and only if for any maximal for (the inclusion) in , we have . Then we can use this definition to be the definition of a saturated couple , in a general (not especially closed under intersection, of course If it is closed under intersection the two definition are equivalent)

With this general definition we can ask ourself if we can say something about the ratio .

Is this bounded if is closed under intersction? same question for a general family, with the second definition…

If the answer is “yes” to one of these questions then, the week Frankl is true – meaning is bouded.

We can also wonder if there is always a such that is empty… We would then have proper Frankl…

But it is not the case : I found a intersection closed family where is not empty for any $x$ of the ground set. I can give this contre example, if someone is interested : an easy generalization of it might give some surprising result, maybe to find a contre example of Frankl itself… I did’nt look too much but a computer should help…

]]>Finding the “parameter” that gives a total order for the improper dice feels within reach. Especially since this parameter must collapse to the same value (or nearly same value) for the proper dice. There is hope that this may even clarify what is going on with the loss of intransitivity with the “rescaled” dice as well.

Your “dream” idea of figuring out a rough simple structure for the intransitive dice (like a circular parameter, or multiple parameters for some higher dimension closed surface) is very alluring. If there is no total order, and it isn’t a random tournament, it just begs to at least try looking for that structure.

To pull out the general structure of the tournament I thought it might help to somehow separate out the piece that allows deviations from perfect balance of beating/losing to dice, thus leaving behind a cleaner/simpler structure. I started playing with it a bit, and have the following in some of my notes:

For the multiset or sequence dice, because of the presence of the involution (let’s denote it n(A) for die A), we can separate any die into a “symmetric” and “asymmetric” piece:

A = (n(A)+A)/2 + (n(A)-A)/2

Counting over all dice, the symmetric part will beat exactly as many dice as it loses to. The asymmetric part will tie all asymmetric parts. So the only thing that can lead to a die deviating from beating exactly as many dice as it loses to, is its asymmetric part compared to the distribution of the symmetric parts of all dice.

The symmetric part has a permutation freedom which satisfies the constraints that the asym part does not. I was hoping that following this could lead to a better view of the general landscape that gives us our tournament. I thought maybe your dream scenario would fit in where we could roughly separate out a “many permutations” structure, and also a kind of “radial” parameter that would ruin the intransitivity if it weren’t for all dice being so “close” to the center on average. So the “radial” parameter would come from the asym part of the die. And ideally this “closeness” concept would coincide nicely with why the strong conjecture fails, and the “radial” parameter would roughly give the positive correlation.

This felt like an interesting possibility for the tournament landscape, but I couldn’t hit upon the right questions to ask or definitions to use to test and explore further.

I regret to say, I never did any tests of your last idea regarding the number of i where a_i > b_i.

Maybe it would be worthwhile to have another post summarizing the remaining questions and how they may fit together with our current understanding. Then gauge how much interest remains in this project. If interest doesn’t return, it would be sad to see the pieces left incomplete, but I guess the open questions could be left as such in the paper to inspire further work.

]]>Does anyone else who participated have a view about what we should do?

]]>(1) has a non empty intersection for all .

(2) has no intersection, for all $k<c$

Let's say that a general family (necessarily with fip and npi)

that satisfy (1) and (2) is a quasi-tower, and let's call the smallest cardinality that can have such a family.

We obviously have .

What seems to be the difficult part in order to prove that

Is it or is it ? Are they both difficult?

If we have one of the two results, we can get rid of conditions that could be difficult to handle with. never the less, there is nothing to lose by having one of the two results…

]]>if there exists such that is greater integer than for all .

Sorry for these notation mistakes! (I'm going to continue these attempt and ideas on mathOverflow, with the same surname, then I will post it here, when I'll be sure that it is correct and relevant!)

I hope I'll get something nice to bring and then bring kind of a compensation for all these inaccuracies, in contrast with the smart behavior and of the high quality of other interventions !

]]>Let’s take a generalization of the problem and figure out when it is simple the cases that we can solve.

Let be a quasi-ordered relation on , such that is thinner than .

For any , we write , or if and . We also write for any , .

We say that has no o-intersection if that it has the o-intersection property (o-ip) if for any finite set , we have not equal to . We say that is o-saturated if it has the oip and no o-intersection. And we say that is a o-total if for any we have or .Let's call resp. ) the smallest ordinal such that there exists a o-saturated set ( resp. o-saturated and o-total) indexed by ordinals lower than .

We can ask our self whether …

We can say that and , and we hope to prove that they are equal.

We can also obviously notice that .

We can also state that the equality holds, for any such that , any has the x-ip. Because we can use the argument of the main comment which is precisely : https://gowers.wordpress.com/2017/09/19/two-infinities-that-are-surprisingly-equal/#comment-203713.

Let's then give an example of such a quasi order. Still calling the increasing injection from , we'll say that if there exists such that for all .

We can see that and , We can

call this (***) then make two statement :

The first one is that $t_{fct}=p_{fct}$.

(because (***) leads that any has the fct.ip. and we just said that the equality holds in that case)

the second statement is that $t_{fct}$ is not smaller than $t=t_f$. Indeed we can then use the argument here : https://gowers.wordpress.com/2017/09/19/two-infinities-that-are-surprisingly-equal/#comment-203534

(***) assure us that the "composition tower" has no fct-intersection witch is stronger then having no f-intersection. For the same reason of strength, we have the obvious opposite inclusion and we then have .

We can also build an intermediate strongly quasi-order that we can call , (for "translation") : if and only if there exists such that . We can see easily that :

(by choosing fine representative element of a tr-tower : = total family indexed by ordinals that extend )

We then have :

lower or equal to lower or equal to (+++)

This make the hope of picking a fct-tower (then a f-tower with the -argument) into a little bit more concrete than before these obervations… because we can suppose that is only when , and is a limit ordinal…

(+++) also show that

is equivelent to

Witch is a translation of the problem in a context of classes closed "up-to translation" (like fct and tr) – note that the composition is compatible with both tr-equivalence and fct-equivalence)

It might also be interesting to get other quasi order, by making an eventually non-empty f-intersection empty by some quotient, and consider a digressing family of such quasi order…

It could also be instructive to build , quasi-order where .

]]>