Τον χειμώνα, όταν έχει παγετό, ψάχνω παπούτσια που να μη γλιστράνε καθόλου.

Τον χειμώνα is an example of the accusative of time in Greek. Greek often uses the accusative, especially with a definite article, to mean during a time period.

So:

• τον χειμώνα = in winter / during the winter
• το καλοκαίρι = in summer
• την Κυριακή = on Sunday

You do not need a preposition here. Using στον χειμώνα would usually sound unnatural in this meaning.

This is a very common Greek pattern with weather and environmental conditions: έχει + noun.

For example:

• έχει κρύο = it is cold
• έχει ζέστη = it is hot
• έχει αέρα = it is windy
• έχει παγετό = there is frost / it’s frosty

So even though English uses it is or there is, Greek often uses has in these expressions. It is idiomatic and very natural.

In expressions like έχει παγετό, the noun is usually used without an article because it refers to a general condition, not a specific frost.

Compare the idea in English:

• There is frost → general weather condition
• There is the frost → would sound specific and unusual

So έχει παγετό is the normal idiomatic way to say there is frost / it’s frosty.

Greek uses the present tense here because it describes a habitual or general action, not just something happening right now.

So the sentence means something like:

• In winter, when there is frost, I look for shoes...
• or ...I look for / tend to look for shoes...

This is the same kind of present tense English uses for habits:

• When it rains, I take an umbrella.
• In winter, I wear boots.

Because in this context Greek is talking about shoes/footwear as the kind of thing the speaker is looking for, not one single shoe.

Just like in English, shoes is normally plural when you mean a pair or footwear in general:

• ψάχνω παπούτσια = I’m looking for shoes
• ψάχνω ένα παπούτσι = I’m looking for one shoe

So the plural is the natural choice here.

This is one of the most important points in the sentence.

που να + subjunctive is often used after nouns when you mean:

• something you are looking for
• something you want
• something suitable
• something with a desired characteristic

So:

• ψάχνω παπούτσια που να μη γλιστράνε
  = I’m looking for shoes that won’t slip / that do not slip

This does not mean you already know which shoes they are. It expresses a desired quality.

If you said:

• ψάχνω τα παπούτσια που δεν γλιστράνε

that would sound more like:

• I’m looking for the shoes that don’t slip

In other words, a more specific set of shoes, as if such shoes already exist and you are trying to find them.

So που να is the better choice for desired suitability.

Because που να is a common structure in Greek for a relative clause that expresses expectation, purpose, suitability, or desired result.

It often appears after verbs like:

• ψάχνω = look for
• θέλω = want
• βρίσκω (in some contexts)
• nouns referring to something needed or suitable

Examples:

• Θέλω ένα μέρος να είναι ήσυχο.
  = I want a place that is quiet.
• Ψάχνω κάποιον να μιλάει αγγλικά.
  = I’m looking for someone who speaks English.
• Χρειάζομαι παπούτσια που να είναι άνετα.
  = I need shoes that are comfortable.

So here που να μη γλιστράνε means that won’t slip / that do not slip, with the idea of a desired feature.

Because after να, Greek uses μη(ν) for negation, not δεν.

So:

• να γλιστράνε = to slip / that they slip
• να μη γλιστράνε = not to slip / that they not slip

This is the normal rule:

• δεν negates the indicative
• μη(ν) negates the subjunctive, imperative-related forms, and certain non-indicative structures

Since να introduces a subjunctive-type clause, μη is required.

Both forms exist.

• γλιστράνε
• γλιστρούν

They both mean they slip / they are slippery / they skid, depending on context.

In Modern Greek, many verbs have two present plural forms like this:

• μιλάνε / μιλούν
• χτυπάνε / χτυπούν
• γλιστράνε / γλιστρούν

The -άνε form is very common in everyday speech and sounds natural here. The -ούν form can sound a bit more formal or simply be an alternative, depending on the verb and speaker.

Καθόλου means at all or not at all.

So:

• να μη γλιστράνε = that don’t slip
• να μη γλιστράνε καθόλου = that don’t slip at all

It strengthens the negation and makes the meaning more emphatic: the speaker wants shoes that are not slippery in the slightest.

You will often see καθόλου with negatives:

• Δεν μου αρέσει καθόλου. = I don’t like it at all.
• Δεν πεινάω καθόλου. = I’m not hungry at all.

Grammatically, yes, but the nuance changes.

• που να μη γλιστράνε καθόλου
  = shoes that won’t slip at all
  → desired property; suitable shoes the speaker wants to find

• που δεν γλιστράνε καθόλου
  = shoes that don’t slip at all
  → more like a factual description of shoes already being identified

So if you are talking about the kind of shoes you are looking for, που να μη γλιστράνε καθόλου is more natural.

Because όταν έχει παγετό is an inserted time clause: when there is frost.

The sentence has this structure:

• Τον χειμώνα,
• όταν έχει παγετό,
• ψάχνω παπούτσια...

The commas help separate the time expressions from the main clause and make the sentence easier to read. In Greek, punctuation with introductory or inserted clauses often works similarly to English.