Söylenti dün internette yayılmış.

I hear that a rumor spread online yesterday. (Secondhand report)

Breakdown of Söylenti dün internette yayılmış.

• dün: yesterday
• internet: the internet
• yayılmak: to spread
• söylenti: a rumor (unverified report in circulation)

Questions & Answers about Söylenti dün internette yayılmış.

What does yayılmış mean exactly, and how is it built?

Yayılmış comes from the verb yayılmak, which means to spread.

It can be broken down like this:

• yay- = spread
• -ıl- = makes it yayılmak, an intransitive/passive-type form meaning to spread rather than to spread something
• -mış = the inferential / reported past

So yayılmış means something like:

• apparently spread
• is said to have spread
• seems to have spread
• sometimes simply has spread, depending on context

In this sentence, Söylenti dün internette yayılmış, the idea is that the rumor spread yesterday on the internet, but the speaker is presenting it as learned indirectly or inferred.

Why is it yayılmış and not yaydı or yayıldı?

These forms express different things:

• yaydı = spread something
  This comes from yaymak, the transitive verb. It needs an object.
  Example: Haberi internette yaydı. = He/she spread the news on the internet.

• yayıldı = spread
  This comes from yayılmak and is the direct past tense.
  Example: Söylenti internette yayıldı. = The rumor spread on the internet.

• yayılmış = apparently spread / is said to have spread
  This is also from yayılmak, but with -mış, which adds the inferential/reported meaning.

So yayılmış is used because the sentence is not just about a past event, but about a past event presented as something the speaker learned indirectly.