The Times Square area of New York City began deteriorating in the 1950s; in the 1970s, the area was a center for pornography and crime. The nickname “Slime Square” (slime + Times Square) was coined in Variety (New York, NY) on June 20, 1956.
 
Times Square was redeveloped in the 1990s and the “Slime Square” nickname is mostly of historical interest.
     
The New York Times newspaper (formerly located in Times Square) is sometimes nicknamed the “New York Slimes.”
     
 
Wikipedia: Times Square
Times Square is a major commercial intersection, tourist destination, entertainment center and neighborhood in the Midtown Manhattan section of New York City at the junction of Broadway and Seventh Avenue. It stretches from West 42nd to West 47th Streets. Brightly adorned with billboards and advertisements, Times Square is sometimes referred to as “The Crossroads of the World”, “The Center of the Universe”, “the heart of The Great White Way”, and the “heart of the world”. One of the world’s busiest pedestrian areas, it is also the hub of the Broadway Theater District and a major center of the world’s entertainment industry. Times Square is one of the world’s most visited tourist attractions, drawing an estimated 50 million visitors annually.
 
20 June 1956, Variety (New York, NY), pg. 1, col. 3:
CONEY & B’WAY: SHABBY TWINS
SODOM-BY-SEA VS. SLIME SQUARE
By Jo Ranson
Two of the great New York City entertainment beats of yesteryear—Coney Island and Broadway—are having a contest to see which is the shabbier. The greasy, vulgar feeling of Surf Avenue is now duplicated on the Main Stem. The Longacre which became Times Square could read Dime Square; and on some nights and corners, substitute Slime Square.
   
17 August 1961, Daily News (New York, NY), pg. 30, col. 1:
West 42d St. at 7th Ave.: Slime Square
By KITTY HANSON
 
23 May 1963, Variety (New York, NY), pg. 2, col. 1:
New York Rabbi Slams
Shabby ‘Slime Square’:
Conditions “Disgraceful’
Times Square, more particularly 42d and 8th Ave., was given a severe back of the hand from the pulpit last Saturday (19) by the Rev. Dr. William F. Rosenblum in a sermon in Manhattan’s Temple Israel. He called the area “a lurid slime centre.”
 
6 November 1968, Variety (New York, NY), pg. 1, col. 1:
Hookers, Homos, Pornos Unchecked,
Under Civil Rights, in ‘Slime Square’
New York’s Times Square, the hub of Manhattan’s show biz and site of famous real estate, has of late often been termed “Slime Square” or “Crime Square” by many who have seen the area slide from respectability to sleaziness to downright tawdriness.
     
Google Books
Double-speak in America
By Mario Pei
New York, NY: Hawthorn Books
1973
Pg. 62:
There is little doubt that all analogical coinages dealing with the deterioration of New York City are of journalistic origin. They include “Times Scare,” “Slime Square,” “Crime Square,” and a “Gun City,” which is obviously based on Mayor Lindsay’s “Fun City.”
 
Google Books
Show Business Laid Bare
By Earl Wilson
New York, NY: New American Library
1975, ©1974
Pg. 134:
The hooker section stretched from the West Forty-second Street area that the late editor Abel Green of Variety named Slime Square northward to about Fifty-second Street. That, in the old, old days, had been the westernmost end of Swing Street.
 
YouTube
Ghostbusters 2 - Slime Square
Ghostbusters.net
Published on Apr 11, 2011
The Ghostbusters attempt to warn the Mayor of the slime invading the city of New York.
   
Gothamist
A Look Back At The Real Deuce: Times Square In The 1970s
BY JEN CARLSON IN ARTS & ENTERTAINMENT ON SEP 8, 2017 1:55 PM
(...)
A year later (in 1972—ed.), Variety called the area “Slime Square,” and the theater district demanded a clean-up.