Which causal structures are absent from any "nice" patch of Minkowski space?

Question

Which "causal separation structures" (or "interval structures") can not be found among the events in "any nice patch ($P$) of Minkowski space"?,

where "causal separation structure" ($s$) should be understood as a function from $n (n - 1) / 2$ distinct pairs (formed from $n$ elements/events of some set $E$) to the set of three possible assignments of "causal separation" (namely either "timelike", or "lightlike", or else "spacelike").

Of particular interest: what's the smallest applicable number $n$? -- for which a corresponding "causal separation structure" can be expressed which can not be found among the events in patch $P$; i.e. such that $E \, {\nsubseteq} P$.

Finally: please indicate under which conditions (on which sort of patches, necessarily different from "any nice patch of Minkowski space") the proposed structure could be found instead; or otherwise, whether it is "impossible" in general.

Answer 1

It seems you don't us to differentiate between past time-like (null) and future time-like (null). Perhaps you have in mind more general spacetimes, but I'm having a hard time imagining a situation where one could distinguish time-like and space-like separated points but not past time-like and future time-like. If one was allowed to differentiate future and past then one would have a simple example with three points.

Without past and future, the minimal impossible configuration can be made with four points. Since any three points which are all lightlike to each other other determine a null ray one cannot have four points ABCD such that:

A, B and C are lightlike separated from each other, B, C and D are all lightlike separated from each other but A and D are spacelike or timelike separated.

Answer 2

One applicable "causal structure" involving 15 events can be illustrated as a subset of all events attributable to "five participants (conveniently called ${\mathcal A}, {\mathcal F}, {\mathcal J}, {\mathcal N}$ and ${\mathcal U}$), each finding coincident pings from the four others".

Of the 15 events to be considered, each of the five participants shares in three events:

${\mathcal A}$ takes part in events A, B and C,

which are (obviously supposed to be) pairwise timelike to each other, and with a consistent (causal, "nice") assignment of past or future direction; similarly

${\mathcal F}$ takes part in events F, G and H,
${\mathcal J}$ takes part in events J, K and L,
${\mathcal N}$ takes part in events N, P and Q, and
${\mathcal U}$ takes part in events U, V and W;

further:
AG, GC, AK, KC, AP, PC, AV, and VC are lightlike,
FB, BH, FK, KH, FP, PH, FV, and VH are lightlike,
JB, BL, JG, GL, JP, PL, JV, and VL are lightlike,
NB, BQ, NG, GQ, NK, KQ, NV, and VQ are lightlike, and
UB, BW, UG, GW, UK, KW, UP, and PW are lightlike;

the separations of all ten pairs among the events A, F, J, N, U are spacelike,
the separations of all ten pairs among the events B, G, K, P, V are spacelike,
the separations of all ten pairs among the events C, H, L, Q, W are spacelike, and finally

the separations of all twenty remaining event pairs are timelike;
all together with consistent/causal "direction assignments".

Here a (sketch of a) proof that this structure can not be found in a patch of Minkowski space (including its "nice/obvious direction assignments"):

(1) In a suitable "projection into 3D-flat (Euclidean) space", but without loss of generality, events G, K, P, V are (supposed to be) situated on the surface of a sphere, at whose center are (in coincidence) events A and C, and with event B inside this sphere (but not necessarily coinciding with A and C). Further

(2) events B, G, K, P are situated on an ellipsoid with focal points U and W; and moreover, at equal distance from U (and also from W). Consequently B, G, K, P are situated on a circle on a plane perpendicular to the ellipsoid axis UW, while event V is inside this ellipsoid.

(Similarly, for events B, G, K, V wrt. to the ellipsoid axis NQ, and so on.)

However: if G, K, P are situated on a sphere, and B, G, K, P are situated on a circle, then B is situated on that sphere as well; in contradiction to (1), q.e.d.

In turn, as far as this argument would fail in a (or even in any) non-Minkowski case, without the possibility of "a suitable projection", the described "structure" is perhaps not ruled out, but may instead be found/present.

p.s. Since the sketch of the proof (as I noticed only after having it written down and submitted) doesn't even explcitly mention the six events F, H, J, L and N, Q at all, the "structure" between the nine remaining events (explicitly mentioned in the sketch of the proof) appears sufficient to carry this particular proof that this "structure" could not be found in Minkowski space; therefore apparently $n \le 9$.