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断ってきた

has refused

→ 断る
phrase Rare #39,551
has refused Often describes refusals made repeatedly or over a period up to the present. Depending on context, てきた can also suggest that someone came back with a refusal.
JA: 彼は何度も誘いを断ってきた。
EN: He has turned down the invitations many times.

その協会は設立以来、素人の入会を断ってきた。

That association has refused to admit amateurs since it was founded.

彼らは招待を断ってきた。

They turned down the invitation.

最後の最後に断ってきた。

They turned me down at the very last minute.

あと払いで買ってほしいと言ってきた人を、ずいぶん断ってきたはずよ

I’m sure I’ve turned down quite a few people who asked me to buy things for them on credit.