The down-set we start with can be replaced by any arbitrary directed graph, and the goal is then to find a map from this directed graph to some undirected graph with the following properties: if we consider the set of elements of that are reachable from , then we want the induced subgraph to be connected, and we want to be of maximum diameter. ]]>

Note that if we consider, for each of the -shadows, the indices at which they appear and then disappear in the sequence (they span intervals so this is well defined), and order them by the interval order (first by when they appear and then by when they disappear), then this forms a convex sequence of -uniform singleton multiset families, whose length is thus bounded by , which means that the number of intervals in the partition is at most .

Then, if we use this generalized Lemma as in the proof of Corollary 1, we have to bound the sum of the supports . If we sum the number of -shadows instead of the sum of the number of -shadows, because of the generalized Lemma 1, we know that this is upper bounded by , but the number of -shadows is larger than the number of -shadows for .

Ideally we would want to show that for we have .

.

With this definition, we obviously have and we can prove easily that and it seems that although I don’t have a proof for it yet.

It would be interesting to study the function defined as the maximum -diameter for down-sets over elements.

]]>- For , this gives a tight bound, which means that the decomposition with respect to 1-shadows is optimal for this value of .
- One interesting aspect of the decomposition is that the supports of the segments, ie the sets are in number at most and satisfy which is effectively .

Can these be generalized somehow?

In particular, can one decompose a convex sequence of 3-uniform multisets with respect to its 2-shadows?

For example for , this sequence does it: (with diameters ). For , this sequence has diameters .

The case is where things are likely becoming interesting since we know that .

I am considering the following equivalent formulation: given a down-set (which is nothing but the union of the since one can always add subsets of the existing elements of to complete it into a down-set), the convexity and disjointness condition corresponds to the existence of a labeling function which is such that for any , maps the set to an interval.

In this formulation, the question is how large can the interval be?

Let’s call this the diameter of , denoted by and defined as .

It seems that the maximal elements of play a crucial role in controlling this diameter. Let’s call the set of maximal elements of . A simple case is when is a singleton, say . Indeed in this case, one can easily prove that (this corresponds to the case where one of the contain which gives in the standard formulation).

Similarly, it seems that when one can prove .

However, the situation becomes quickly tricky when one considers three elements.

{11111, 11112, 11123, 11234, 12345, 23455, 34555, 45555, 55555}

In this case, if you consider all the subsequences whose support contain one particular element, summing up Nikolai’s bound for them (ie if you consider that there length is bounded by f^*(d-1,k) for some appropriate k) gives a total length that is always larger than f^*(5,5), no matter how the sequence is decomposed. So this decomposition approach does not seem to allow to prove Nikolai’s conjecture but only a weaker upper bound. ]]>

It seems that this is the case (on some simple examples I looked at), provided one can construct the sequence without being forced to start at one end. So in other words, we need to find a decomposition which minimizes , and on some examples this seems to be possible and yield the desired properties.

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