Combine the cucumber, onion, red pepper and salt in a large bowl. Cover with a clean kitchen towel and then layer ice cubes on top of the towel. The ice helps to make the pickles crisp. Refrigerate for 4 hours or overnight. When you're ready to make the pickles, discard the ice and rinse the cucumber, onion and red pepper. Drain well and rinse again. Drain well.
Bring the remaining ingredients to a boil in a stockpot, stirring to dissolve the sugar. Once at a boil, add the cucumber, onion and red pepper to the pot and bring the brine back to a boil. As soon as it starts to boil again, start packing the pickles into clean (and sterilized if you plan on keeping the pickles for any length of time) glass jars. Leave about an inch of room in the top of each jar. Fill the jars with the remaining brine so that there is only 1/2 inch of room left. Cover with clean (and sterilized) jar tops and secure with the screw lids.
If you want to keep the pickles out of the fridge, you will need to process the jars now. Bring a large pot of water to a boil and place a rack in the bottom of the pot. Once at a boil, add the jars to the pot, making sure they are resting on the rack and not the bottom of the pot, and making sure there is at least 1 inch of water above the jars. Boil for 10 minutes and then remove. As the jars cool, you should hear a little pop at the lids seal. If the lid tops are not sucked down, it means they did not seal properly. Store these jars in the refrigerator. Otherwise, store the pickles at room temperature.
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