
1) Throw down the gauntlet – бросить кому-то вызов (“gauntlet” – защитная перчатка). Как можно догадаться, “ pick up/take the gauntlet” означает “принять вызов”.
He has thrown down the gauntlet to his competitors.
2) Look to one’s laurels – пытаться удержать свое первенство, а не просто почивать на лаврах.
With customers becoming more and more demanding and technological progress being so rapid, the older, established companies are having to look to their laurels.
3) Snap at one’s heels/nip on one's heels – наступать кому-то на пятки.
Leisure activities connected to virtual reality has been nipping on the heels of the traditional ways of spending pastime.
4) Pit one’s wits – состязаться в эрудированности.
I could never pit my wits against my dad, he was a real bookworm.
5) Dog-eat-dog – ситуация, в которой люди готовы пойти на все ради победы.
Model business is a dog-eat-dog world.
Cheers!