What to text to ask her out after a good back and forth

Five or six good messages is a count, not a signal. What you actually want to check is whether she's been bringing things into the conversation herself. If she has asked you questions, picked up on something you said, or offered her own stories without being prompted, she's invested. If every reply is a short answer to your question, she may still be interested but she's waiting for you to drive. Pay attention to the topics she's opened. In your example she mentioned ramen and called out a specific place. That's an opening, because she has given you a concrete thing you can build an ask around.

Also look at her pacing. If she's replying within a few hours and her messages are getting longer, that's a good sign. If the replies are getting shorter or taking a day or more, she may be less keen, and a direct ask is a more honest test than another round of small talk. You don't need to read minds. You need enough evidence to justify a single clear question.

The ask works best when it names a thing, a place or an activity, and a time window she can say yes or no to. Tying it to something she already said does most of the work. If she brought up a ramen place, the ask is about that place, not a vague coffee or drink. Give one day or a rough range, then stop. Offering too many options makes her choose and makes the answer feel like a negotiation. Keep the message short enough that the easiest response is a yes, a no, or a counter-day.

You should also make it clear you'd be fine with a no. You don't have to say it, but the tone should not depend on her saying yes. Write it like you'd be happy either way. People say yes more often to a plan that doesn't feel like a test.

Give her a full day before following up, more if she usually replies slowly. A second message the same evening reads as pressure even if you don't mean it that way. When you do follow up, keep it light and don't repeat the full ask. Reference the plan again briefly, or offer a different day, so she has an easy way back in. If she still doesn't answer after one follow-up, take the silence as the answer. Don't send a third message asking whether she got the first one. Let it sit. If she comes back later with a warm message, you can pick the conversation up again without making it a big deal.

The point of one follow-up is to give her a second chance to answer. Beyond that, you're no longer giving her space, you're asking her to respond on your schedule.