Main paradoxon of archaeogenetics:

After a couple of millenia, every human is related.

Proof:

• Every person has two biological parents
• All parents had two parents
• The average age of mothers at birth was ⋍20-33 years
• Each century yields about 3-5 generations
• There are 20 centuries since Caesear
• Therefore there are ⋍ 60-80 generations since 0BC
• Therefore you have roughly 2 to the 60-80 combinatorial anchestors going back two millenia ⋍ 10²⁴
• The "total number of people who have ever lived" is in the order of 100 billion ⋍ 10^11
• Therefore to come to 10²⁴ combinatorial anchestors, the vast mayority of parents need to appear in several paths

If nothing else, this should give a sense of how incredibly dense the forest (network!) of anchestry becomes after two millenia. Going back 2000 years one has about as many anchestors as there are grains of sand in the Sahara.

Induction

Europe

"Every one is a descendent of Charlemagne"

The argument has been made for the whole of Europe by others in more detail

The specific choice of Charlemagne is a bit misleading since it may suggest that his high procreativity leads to the property of all-relatedness, which is entirely false: Any reproduction rate bigger than zero (which does not fizzle out) leads to this property: If after some generations one member of and inbreeding small community mates with an outsider, all offspring naturally inhabit the larger anchestry tree of the outsider; and so on.

"By 1,000 years ago everyone (who left descendants) would be an ancestor of every present-day European."
This is the correct way of phrasing the above observations.

India and China

In order to spread the all-relatedness property to remote places, it suffices to note that this utterly comprehensive gene-flow can jump from one population / gene-pool to another with a single newcomer / outsider halfway. So if by 1000AD "Every single 'native' human in Europe was related to Caesar", then one of these Europeans traveling to China/India/America (and breeding) set off the same chain reaction: This newcomer acts as the new local Caesear. In practice there were already many Caesears in India/China before 1000AD and certainly many travelers after.

America

For the numbers to be large enough to spread all over America, the 500 years since Columbus might be a bit terse (this is an excercise), however if one accepts the likelyhood of one single outsider reaching America around 1000BC, the argument is obvious. (Lost sailor/ undocumented Chinese / Viking expedition etc).

Sentinel Islands

Let us investigate the most remote places on earth:
For smaller populations like the Sentinel Islands, Pygmys, Amazons, New Zealand etc one doesn't need to assume early newcomers but a relatively recent outsider interbreding suffices (this is an excercise)

Note if you are only interested in the outmost branches of the paths the same estimates hold because

• 2^n = 1+∑2^i for (i<n) (16=1+8+4+2+1)

It is a paradoxon because

1. It is beyound what most people believe
2. It may be ill-formulated because with low probability there could be a handful (1000?) exceptions
3. It is entirely plausible (unless you find a fundamental flaw in reasoning)
4. Thus it is entirely debatable (and thank god there are fewer cocktail parties these days)

Objections

a) People in some communities may mostly marry first cousins.
Response: As long as one individual per epoche breeds with an outsider, the continent argument from above applies.

Follow up questions:
Does this make the whole concept of races irrelevant?
Yes and no: male y-chromosome lineages are a read thread, even if the whole genome contribution might be 1/10²⁴
Modern concepts of races should be more like a flux balance of clusters in PCA charts.

Corollaries:

anchestors without gene

parents-without-a-trace-paradoxon: With 10²⁴ combinatorial anchestors, some of them might not have left a single base pair in your DNA, which has only about 6,2 × 10^9 base pairs.

caesar-paradoxon

Every living human is related to Caesar (implicit assumtion: successful leaders are very likely to procreate)

aborigines-paradoxon

Every living human is related to all Australian aborigines living 2000 years ago.

The Web of Life

This bigger picture of

https://aeon.co/essays/why-evolution-is-not-a-tree-of-life-but-a-fuzzy-network

Quotes:
«Does data support the hypothesis that species don’t just separate, they also merge? The answer is a resounding yes.»
«that process of splitting up and merging back together again, and getting a bit of DNA from here to there, that’s happening all the time, in all of the tree of life»
‘Different pieces of DNA have different gene trees, and you just can’t represent all of that in a single diagram.’

5% - 8% of human genome appears to be from [retro]virus inserts. Thats outside the tree of descent.