The organisers of Swiss fair Art Basel have filed a lawsuit against Adidas for using the event's name and branding without permission on a limited-edition pair of trainers.

The 24-page lawsuit, filed in Florida on 30 May, argues trademark infringement in reference to a pair of trainers Adidas presented at the Design Miami fair last year.

According to the art fair's parent company, MCH Swiss Exhibition Limited, Adidas didn't seek permission to use the fair's name before stitching it into the tongues of 500 pairs of its EQT trainers.

And although Adidas didn't sell the shoes – they were given away for free after a performance – the sneakers have been sold online through eBay by those who attended the event.

Art Basel believes the turn of events "dilutes" its trademark. The lawsuit seeks to stop Adidas from using the name and instructs the company to destroy all remaining shoes.

"The infringing sneakers have been, and continue to be, offered for sale and sold on eBay," said Art Basel's Miami lawyer, Joseph Englander, in the document. "The suit seeks to stop Adidas from using the Art Basel name, and to destroy all remaining shoes — and to pay damages for diluting the trademark."

Dezeen reached out to both Adidas and Art Basel for comment, and while a spokesperson for the fair said "we feel the complaint speaks for itself", Adidas is yet to respond.

Design Miami, a high-end collectable design fair, takes place in Miami Beach every December. Its sister event, Design Miami/Basel, coincides with Art Basel every June.

In the claim, the fair's organisers discuss the value of the brand as part of its argument, claiming the "estimated value of the art brought to Art Basel in 2016 was over $3 billion".

They go on to talk about its "unsolicited media coverage" before accusing Adidas of contacting journalists to write about the shoe, with the filing providing links to four online websites for sneaker enthusiasts.

The 2017 edition of Art Basel and Design Miami/Basel due to take place next week. Last year's events saw Jean Prouvé's demountable office restored and used as a display and a floral Zaha Hadid tribute at the Vitra Fire Station by the Bouroullec brothers.