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I'm flying from UK to Germany with Easyjet (there's next to no weight restriction to hand luggage). I've never had a problem with various edible items before, on any of my EU routes. I don't see how a watermelon would be different from a plastic bag with 10 sandwiches. But still, can I have a whole watermelon in my hand luggage ? What about a half melon ?

There are two potential issues here: Airport security and customs.

For airport security:
In the UK, "Liquids include liquid or semi-liquid foods, eg soup, jam, honey and syrups" and "Liquids in containers larger than 100ml generally can’t go through security even if the container is only part full." Exemptions (e.g. essential medical purposes) don't seem likely to apply here.

Watermelon is less dense than honey, so an agent might use that to classify it more toward the "liquid/container" side than the "solid" side of honey, and honey (>100 mL) is not allowed. It is also less viscous than some honey. Edit to clarify: criteria used in practice for classification is a different question than that of what more scientifically-based criteria agents perhaps should use.

In the US, the TSA lists it as approved for carry-on in their "Can I Bring" app. Although this is the result for the explicit query "watermelon," the response does have a disclaimer that "Liquid or gel food items larger than 3.4 oz are not permitted in carry-on baggage" so if an agent decides that a watermelon is a non-empty container capable of holding 3.4oz of liquid or gel, they might use their wide discretion to tell you it's not allowed.

For customs:

Although generally speaking when entering Germany, "The import of food and animal feed for the consignee’s private use or consumption is permitted in principle," other readers traveling to other countries should consider whether or not they're allowed to transport fruits and seeds across national borders. Rules are much less restrictive for travel within the EU.

While I am unaware of any liquid list containing a watermelon, here's the biggest list of liquids I am aware of, from Debrecen airport:

All drinks (including beverages, water, soup, syrups), chocolate creams, creams, oil, cheese spread, liver cream, pastas, peanut butter, yogurts, cottage cheese, butter, margarine, frozen food, tins, liver wurst, sausage, jelly, mousse, hair colorants, mascara, fluid lipsticks, fluid masks, lotions, spray and roll-on deodorants, perfumes, nail polish removers, and all items made of similar substances.

Of course, there's nothing to guarantee the UK will use the same list but knowing how the UK is batshit crazy about airport security I bet their list is even worse. If liver wurst and sausage is a no-go then you can bet a watermelon won't pass either.

Watermelons can easily be loaded up with other liquids including potently explosive liquids and as such would likely be prohibited as carry on items. Perhaps folks in the UK would not be familiar with this aspect, but in parts of the USA the process of spiking watermelons with a bottle or two of whiskey is commonly done to get alcohol into venues like outdoor concerts, sporting events, where alcohol is otherwise banned.

But one has to ask, are the watermelons that good in the UK?

Sharing my personal experience on the flight from Venice to Gdansk with three legs and thus three security checks (Ryanair uses cheap small airports where you can't transfer) in Treviso (Italy), Beauvais (France) and Modlin (Poland). I have got a small watermelon (15cm in diameter) in my backpack, which I took as a cabin baggage. During baggage scans in Italy and France I was asked to open my backpack and security officers checked it manually. Once they found the big berry, they smiled, started joking with their colleagues and let me proceed. I was asked no questions and had no additional checks in Poland.

P.S. I like watermelons :)