Der er flere rene kaffefiltre i skabet, så vi løber ikke tør i denne uge.

Der er is the standard Danish way to say there is / there are when you introduce the existence of something.

So:

• Der er et kaffefilter = there is a coffee filter
• Der er flere kaffefiltre = there are several/more coffee filters

Here, der does not really mean a physical there. It is an existential there, just like in English there are.

A learner often compares:

• Der er flere rene kaffefiltre i skabet = there are several clean coffee filters in the cupboard
• Det er et rent kaffefilter = it is a clean coffee filter

So der er introduces that something exists, while det er identifies or describes something.

In this sentence, flere means several / more than one.

So Der er flere rene kaffefiltre i skabet means there are several clean coffee filters in the cupboard.

A useful thing to know is that flere can mean slightly different things depending on context:

• Jeg vil have flere = I want more
• Flere mennesker kom = several/more people came

Here, since we are just stating what is available, flere is best understood as several.

Because rene agrees with a plural noun.

The noun is kaffefiltre, which is plural, so the adjective takes the plural form:

• et rent kaffefilter = a clean coffee filter
• rene kaffefiltre = clean coffee filters

A quick pattern:

• common singular indefinite: often - form, e.g. en ren kop
• neuter singular indefinite: often -t, e.g. et rent filter
• plural and definite: often -e, e.g. rene filtre, det rene filter

So rene is exactly what you expect before a plural noun.

Because Danish, like German, very often forms compound nouns by writing them as a single word.

So:

• kaffe = coffee
• filter = filter
• kaffefilter = coffee filter

The plural is:

• et kaffefilter
• flere kaffefiltre

English sometimes writes similar expressions as two words, but Danish usually prefers one compound word.

I skabet means in the cupboard / cabinet.

Two things are happening here:

1. i means in
2. skabet is the definite form of skab:
   • et skab = a cupboard / cabinet
   • skabet = the cupboard / cabinet

So:

• i et skab = in a cupboard
• i skabet = in the cupboard

Danish often adds definiteness directly to the noun with an ending, instead of using a separate word like the.

Here så means so or therefore.

It connects the two ideas:

• there are several clean coffee filters in the cupboard,
• so we will not run out this week.

So the structure is:

Der er flere rene kaffefiltre i skabet, så vi løber ikke tør i denne uge.

It is a very common coordinating conjunction in everyday Danish.

Yes, it is basically an idiomatic expression.

at løbe tør means to run out.

Literally, løbe means run, and tør normally means dry, so word-for-word it looks like run dry. But in actual usage, it means that your supply of something is used up.

Examples:

• Vi løber tør for kaffe = we run out of coffee
• Han løb tør for benzin = he ran out of petrol/gas

In your sentence, the object is understood from context: coffee filters. So Danish does not need to repeat for kaffefiltre.

Because in a normal main clause, Danish usually places ikke after the finite verb.

So:

• vi løber ikke tør = we do not run out

This is standard Danish word order in main clauses:

• subject + finite verb + ikke
  • rest

Examples:

• Jeg forstår ikke = I do not understand
• Hun kommer ikke i morgen = she is not coming tomorrow

So vi løber ikke tør is completely regular.

Both can exist in Danish, but i denne uge is very natural when you mean during this week.

• i denne uge = in/during this week

The preposition i helps express the time period more clearly.

Also, denne means this:

• denne uge = this week

You may also hear:

• i den her uge = in this week / this week

That is a more conversational alternative. Denne is a bit more formal or neutral in writing.

Because the noun phrase is indefinite plural.

In English, we can say:

• clean coffee filters
• some clean coffee filters
• several clean coffee filters

Danish works similarly. You do not need an article before an indefinite plural noun phrase.

So:

• rene kaffefiltre = clean coffee filters
• flere rene kaffefiltre = several/more clean coffee filters

If it were definite, it would look different:

• de rene kaffefiltre = the clean coffee filters

Yes, absolutely.

That would be a more explicit version:

• Vi løber ikke tør for kaffefiltre i denne uge = We will not run out of coffee filters this week

In your sentence, for kaffefiltre is omitted because it is obvious from context. Danish often leaves out repeated information when the meaning is already clear.

So both are natural:

• vi løber ikke tør i denne uge
• vi løber ikke tør for kaffefiltre i denne uge

The shorter version sounds efficient and natural when everyone already knows what you are talking about.