No tax to pay (until you reach a ~$4M lifetime limit), but tax papers (Form 709) should be filed in most cases, by the giver, for amounts over $15,000 per year to any person . "In general. If you are a citizen or resident of the United States, you must file a gift tax return (whether or not any tax is ultimately due) in the following situations.
If you gave gifts to someone in 2018 totaling more than $15,000 (other than to your spouse), you probably must file Form 709 ..." https://www.irs.gov/forms-pubs/about-form-709
If you pay your child's tuition, and you pay it directly to the school, there is no need for any gift tax reporting. The school must be a qualifying educational institution.
This exclusion applies to tuition only. Gifts for any other purpose, such as room and board, must be reported if the amount exceeds $15000.
Also, the lifetime gift tax exclusion is now $11.4 million. That means you don't actually have to pay a gift tax until your lifetime gifts exceed that total. But you have to report gifts to an individual that exceed $15000.
Finally, gifts to an individual are not deductible by you, nor are they taxable to the recipient.