How to stop incoming text messages?

I'm travelling overseas and but I still want to use my iPhone 5 for internet. I turned on airplane mode thinking this would solve my problem of not wanting to be charged for roaming. Phone calls are disabled, and I thought texts would be too but I've gotten three messages so far. I haven't checked them but my phone says I'm not allowed until I disable airplane mode. But I'm still worried because the texts got through anyway.

Try turning on the "Do Not Disturb" feature on your iPhone 5. If that does not work, try disabling the notifications in the notification center for Messages.

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