Ahhoz képest, hogy vastag a talp, könnyű a bögre.

As a whole, ahhoz képest, hogy is a very common expression meaning something like considering that, given that, or for something where.

In this sentence, it introduces a contrast between what you would expect and what is actually true:

• vastag a talp → the base is thick
• so you might expect the mug to be heavy
• but könnyű a bögre → the mug is light

So the phrase adds the sense of surprisingly, in view of that fact.

Because Hungarian normally leaves out van in the present tense, 3rd person when the predicate is a noun or adjective.

So:

• vastag a talp = the base is thick
• könnyű a bögre = the mug is light

This is completely normal Hungarian.

You would use van in other situations, for example:

• A bögre könnyű volt. = The mug was light.
• Könnyű vagyok. = I am light.

So there is no missing word here; this is the standard structure.

Ahhoz is the demonstrative pronoun az with the ending -hoz/-hez/-höz, which here gives the idea to that / compared to that.

With képest, Hungarian uses this pattern:

• valamihez képest = compared to something
• ehhez képest = compared to this
• ahhoz képest = compared to that

So in ahhoz képest, hogy..., the literal structure is roughly compared to that, that..., but in real English you translate it more naturally as considering that...

Both are possible, but they are not exactly identical in feel.

• Vastag a talp puts emphasis on vastag
• A talp vastag starts from the base as the topic

Hungarian word order is strongly connected to information structure. In this sentence, the speaker is highlighting the property:

• vastag = thick
• könnyű = light

That is why the adjective comes first in both clauses.

So:

• Vastag a talp, könnyű a bögre sounds descriptive and natural
• A talp vastag, a bögre könnyű is also grammatical, but flatter and more topic-based

Yes, talp often means sole as in the sole of a foot or shoe, but its broader meaning is the bottom/base/supporting part of something.

So with an object like a mug, talp means the base or bottom part.

Here it does not mean a body-part sole. It means the part of the mug that forms its bottom/base.

That is a very natural question.

• a talp = the base
• a talpa = its base

If you want to state the possession more explicitly, talpa is possible, especially if it is already clear what object it belongs to.

For example:

• Vastag a bögre talpa. = The mug’s base is thick.
• Vastag a talpa. = Its base is thick.

In your sentence, a talp sounds a bit more general, as if the speaker is talking about the base as the relevant part under discussion. Hungarian often allows this when the context already makes the relationship clear.

So a talpa would be more explicit, but a talp is understandable and natural in the right context.

Because hogy introduces a subordinate clause, and Hungarian normally separates subordinate clauses with a comma.

So:

• Ahhoz képest, hogy vastag a talp, könnyű a bögre.

The clause after hogy is the background fact:

• hogy vastag a talp = that the base is thick

And the main clause is:

• könnyű a bögre = the mug is light

The comma is standard punctuation here.

Yes, absolutely.

That version is very natural too:

• A bögre könnyű ahhoz képest, hogy vastag a talp.

The difference is mainly in emphasis.

• Ahhoz képest, hogy..., könnyű a bögre puts the contrast frame first
• A bögre könnyű ahhoz képest, hogy... starts with the mug

Both are grammatical. Hungarian often allows this kind of movement depending on what the speaker wants to foreground.

It can mean both.

• könnyű feladat = easy task
• könnyű doboz = light box

In this sentence, because it is describing a mug and its thick base, könnyű clearly means light in weight, not easy.

So the context tells you which meaning is intended.

Not quite.

• bár means although
• ahhoz képest, hogy means considering that / given that / for something that...

They are similar because both can introduce contrast, but ahhoz képest, hogy often carries a sense of surprise relative to expectation.

Compare:

• Bár vastag a talp, könnyű a bögre.
  = Although the base is thick, the mug is light.

• Ahhoz képest, hogy vastag a talp, könnyű a bögre.
  = Considering that the base is thick, the mug is light.

The second version more strongly suggests: you would expect it to be heavier, but it isn’t.