When Ricky (Desi Arnaz) is trying to impress Mr. Murdoch with his show business scrapbook, he says that his first show was "Too Many Girls". This was, in fact, the name of the first and only Broadway show that Desi Arnaz was in, which ran on Broadway from Oct. 18, 1939 - May 18, 1940. It had music by Richard Rodgers and Lorenz Hart. The charming, dashing young Cuban and this show are partially credited with starting a Latin dance craze of "Conga" dancing in the USA at that time. Arnaz was also brought to Hollywood to make the film version at RKO, Too Many Girls (1940), which starred Lucille Ball. This was how the two met.
At the end, Ricky receives a special delivery letter from Hollywood. He tells Lucy, "they want to test me for the lead!" But the only screen test Ricky ever does occurs much later in the series, in Ricky's Screen Test (1954).