These sheets are perfect. The elastic band is wide and strong. We have an adjustable tempurpedic bed that’s essentially made of 2 Twin XL mattresses. It’s not especially thick so standard cal king sheets were a bit too loose and got wrinkly and loose after one night of sleep.These sheets don’t get loose after one night! Because they’re 2 twin xl bottom sheets, and because they have that awesome side elastic band, they stay perfectly taught until you remove them. Even when I raise and lower the head or feet of the bed - these sheets stay in place and don’t come loose!These sheets are definitely cotton and just get softer every time I wash them. I want to buy more sets like this. We got the white on white striped ones. I researched a lot before buying these because I didn’t want to end up with microfiber or polyester sheets - I’ve found those kinds of sheets to be hot and the texture is kinda weird. Well I can attest that these are nice real cotton sheets.And I can also say they stay nice and cool. We live in a very hot place where the outdoor temp is 100+ and the summer is long. We don’t even use a blanket most of the year because the tempurpedic bed is awfully hot (won’t ever buy one again). So it’s vital that our sheets be nice breathable cotton. These sheets are perfect for us. When it gets cold out I’m sure they’ll stay nice and warm under our light blanket.Also - no they are not wrinkle free. If you get them out of the dryer when they are warm they will flatten nicely and be very smooth. Cotton sheets wrinkle - that is 100% unavoidable. But these really don’t wrinkle badly if I smooth them while warm. I love the wrinkles because it is yet another proof that these are made of natural fiber.