The article below is taken from the Urban Gardening Ideas website and gives wonderful details on size of pots to use, and especially how to support your melons as they grow.
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You need to choose a pot that will be large enough for your container watermelon to thrive. Watermelons grow rapidly and require plenty of water, so it is recommended that you go with a 5-gallon or larger size container. Make sure that the container you will be growing watermelons in has enough drainage holes.
Fill the watermelon container with potting soil or other soilless mix. Do not use dirt from your garden. This will compact quickly in the container and will make growing watermelons in containers difficult.
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To read the full article, head over to the Urban Gardening Ideas website.