Aloha Gadi, In your sentence — “Would you like to suggest some options for next week?” — “Next week” is correct. You use “next week” when you're talking about the upcoming week from today. It’s simple, direct, and commonly used when making plans. On the other hand, “the next week” is usually used when you're describing what happened after a specific event — like in a story or report. For example: “She started her new job, and for the next week, she trained with the manager.” So for making plans or scheduling, always go with “next week.” Hope that helps! Let me know if you want help with more English tips like this or want to learn more about Hawaiian English if you want to learn about Hawaii A hui hou - until we meet again.

In general, these relative time phrases with 'next' don't take 'the'. 'Next week', 'next year,', next May', 'next time' - in the same way as 'last' ('last week', 'last Monday', etc). But they do need the article when you're talking about something specific. 'The next time I see you, I'll be married.' 'The next slot the surgery has available is on Monday.' 'The next week you have off is in June.' 'The last time she saw him, he was looking ill.' In your example, both are possible depending on context. If you say 'the next week', it's likely you've been looking at a succession of weeks, for instance. But if it's just a general relative expression of time, 'next week'.