The official anthology of readings from the works of the Gaon of Vilna is entitled "Even Shleima". The choice of the title is based on the acronym of the phrase: the word "even" representing "Eliyahu ben", and the word "shleima" being read as "Shlomo", which was the name of the Gaon: Eliyahu ben Shlomo. The background for the acronym is not that well known.
In the late 1930's Rabbi Shmuel Maltzahn published a manuscript written by Rabbi Chaim of Volozhin (a well known student of the Gaon) where he related the following story: the Gaon was fond of stating that if we look carefully into the Chumash, we will discover allusions to anything and everything that happened, both to the Jewish nation as well as to individuals, even in the centuries following the time of Moshe Rabbeinu. On one occasion Reb Chaim asked of the Gra, "Where is there an allusion to the rebbe?" The Gaon immediately opened a Chumash to parshat Ki Tetzei and reviewed it a bit until he noticed that the phrase "even shleima" was an allusion to his name. Reb Chaim asked the Gra why he chose to search specifically in parshat Ki Tetzei? Whereupon the Gra responded that in Chumash D'vorim there are ten sedrot, and even though we usually count eleven sedrot in D'vorim, Nitzavim and Vayelech they are really one sedrah, "as is known". Each of these ten sedrot corresponds to each of the ten centuries of the "sixth millennium".
The rabbis of the Talmud had a tradition that the world would last (in its present state) for six thousand years, and then Shabbos would occur. The tradition further relays the historical break-down of these six thousand years. The first two thousand years are labeled as "Tohu", since the notion of monotheism was not being popularized. According to the records in Chumash B'reishit, Avrohom Avinu was born in the year 1948 (after the creation), and according to the tradition he began preaching monotheism at the age of 52, which would correspond to the year 2000. This introduced the second unit of 2000 years which represented the two millennia of "Torah development", which ended in the year 4000, which corresponds to the year 240 in the common calendar. Rabbi Yehudah Hanasi, who edited the Mishnayot, passed away in the year 220, so this cut-off point of 240 seems to correspond to the end of the period of the Tanaim. The next 2000-year unit is labeled as "the days of mashiach", as mashiach may come at any date following the year 240. Rabbi Akiva who thought that Bar Kochva might turn out to be the messiah was not aware of this oral tradition, and his contemporaries argued with him saying, that he (R. Akiva) would be dead and buried, and grass would be growing on his grave, and the time for mashiach will not have arrived, because their tradition had it that the mashiach must arrive at some time during the last 2000 years of the 6000-year program for the world.
So according to the Talmudic tradition, the sixth millennium began with the year 1240. At the time Reb Chaim asked of the Gra for an allusion to him, it was after 1740. So the Gra explained that whatever occurred in the first century of the sixth millennium (between the years 1240 - 1340) should be alluded to in parshat Dvorim; and whatever occurred during the second century (between the years of 1340 - 1440) should be alluded to in parshat Va'etchanan, etc. Since the Gra was then in the sixth century (of the sixth millennium), he opened right away to parshat Ki Tetzei, the sixth sedra in Chumash D'vorim.
Rabbi Maltzahn who printed this manuscript of R. Chaim of Volozhin in his sefer "EmmunahV'Hashgachaon" added the following comment: Parshat Ki Tavo is the seventh sedrah, and should contain allusions to things that would occur between the years 1840 - 1940. At the end of the 1930's the Nazis had already begun their extermination of the Jews, and he suggested that perhaps the bitter "Tochacha" that appears in Ki Tavo was not only an allusion to the many years of suffering of the galut, but also specifically alluding to the Nazi persecutions at the end of the 1930's.
In later years, others pointed out that in the next- the eight sedra, parshat Nitzavim-Vayelech which really is only "one sedra" "as is known"; we read of the return of the Jewish people to Eretz Yisroel, the great teshuvah movement, and the mitzvah of writing a sefer Torah. All of these were witnesses following the year 1940, through the establishment of Medinat Yisrael, the great world-wide baal teshuvah movement, and the popularity of the mitzvah of writing a sefer Torah.
We hope that Hashem will be "mosif machol al hakodesh" and will be "mekabel shabbos", and hasten the coming of the mashiach before the end of the sixth millennium.