Shavuot: the holiday without a date
כדתני רב שמעיה עצרת פעמים ה' פעמים ששה פעמים שבעה הא כיצד שניהן מלאין חמשה שניהן חסרין שבעה אחד מלא ואחד חסר ששה
The Gemara answers: As Rav Shemaya taught in a baraita: Shavuot sometimes occurs on the fifth of Sivan, sometimes on the sixth of Sivan, and sometimes on the seventh. How so? If both the months of Nisan and Iyyar are full months of thirty days each, the festival of Shavuot, which is celebrated fifty days after the second day of Passover, occurs on the fifth of Sivan; if both months are short, with twenty-nine days each, it occurs on the seventh of Sivan; and if one of them is full and the other is short, it occurs on the sixth of Sivan. According to this opinion, if both months were full and the festival of Shavuot occurred on the fifth day of Sivan, and one made his vow on the day after Shavuot, i.e., the sixth of Sivan, and in the following year both months were short, so that the festival of Shavuot occurred on the seventh of Sivan, a whole year would have passed without three Festivals.

Lubavitcher Rebbe, Likutei Sichos 3 p945, https://www.chabad.org/therebbe/article_cdo/aid/93594/jewish/Shavuos.htm

With regard to the Counting of the Omer, however, the crossing of the dateline makes a significant difference. As mentioned above, the counting of the Omer is a mitzvah which is dependent on every person as an individual. Thus when a person crosses the dateline in the middle of counting the Omer, he must continue according to his own personal reckoning although everyone around him is counting a different day.

For example, Pesach falls on Shabbos. On Monday, the second day of the Counting of the Omer , a person travels from east to west [e.g., from the U.S. to Australia]. Although he left on Monday, when he crosses the dateline, it will be Tuesday. That night [the night between Tuesday and Wednesday], he is required to count the third day of the Omer, while the local people will be counting the fourth day.

Conversely, if a person crosses the dateline while traveling from west to east, leaving Monday and arriving on Monday, on the night between Monday and Tuesday, he must count the third day of the Omer, although the local people will be counting the second day.

(טו) וּסְפַרְתֶּ֤ם לָכֶם֙ מִמָּחֳרַ֣ת הַשַּׁבָּ֔ת מִיּוֹם֙ הֲבִ֣יאֲכֶ֔ם אֶת־עֹ֖מֶר הַתְּנוּפָ֑ה שֶׁ֥בַע שַׁבָּת֖וֹת תְּמִימֹ֥ת תִּהְיֶֽינָה׃ (טז) עַ֣ד מִֽמָּחֳרַ֤ת הַשַּׁבָּת֙ הַשְּׁבִיעִ֔ת תִּסְפְּר֖וּ חֲמִשִּׁ֣ים י֑וֹם וְהִקְרַבְתֶּ֛ם מִנְחָ֥ה חֲדָשָׁ֖ה לַיהוָֽה׃
(15) And from the day on which you bring the sheaf of elevation offering—the day after the sabbath—you shall count off seven weeks. They must be complete: (16) you must count until the day after the seventh week—fifty days; then you shall bring an offering of new grain to the LORD.
(א) בַּחֹ֙דֶשׁ֙ הַשְּׁלִישִׁ֔י לְצֵ֥את בְּנֵי־יִשְׂרָאֵ֖ל מֵאֶ֣רֶץ מִצְרָ֑יִם בַּיּ֣וֹם הַזֶּ֔ה בָּ֖אוּ מִדְבַּ֥ר סִינָֽי׃ (ב) וַיִּסְע֣וּ מֵרְפִידִ֗ים וַיָּבֹ֙אוּ֙ מִדְבַּ֣ר סִינַ֔י וַֽיַּחֲנ֖וּ בַּמִּדְבָּ֑ר וַיִּֽחַן־שָׁ֥ם יִשְׂרָאֵ֖ל נֶ֥גֶד הָהָֽר׃ (ג) וּמֹשֶׁ֥ה עָלָ֖ה אֶל־הָאֱלֹהִ֑ים וַיִּקְרָ֨א אֵלָ֤יו יְהוָה֙ מִן־הָהָ֣ר לֵאמֹ֔ר כֹּ֤ה תֹאמַר֙ לְבֵ֣ית יַעֲקֹ֔ב וְתַגֵּ֖יד לִבְנֵ֥י יִשְׂרָאֵֽל׃ (ד) אַתֶּ֣ם רְאִיתֶ֔ם אֲשֶׁ֥ר עָשִׂ֖יתִי לְמִצְרָ֑יִם וָאֶשָּׂ֤א אֶתְכֶם֙ עַל־כַּנְפֵ֣י נְשָׁרִ֔ים וָאָבִ֥א אֶתְכֶ֖ם אֵלָֽי׃ (ה) וְעַתָּ֗ה אִם־שָׁמ֤וֹעַ תִּשְׁמְעוּ֙ בְּקֹלִ֔י וּשְׁמַרְתֶּ֖ם אֶת־בְּרִיתִ֑י וִהְיִ֨יתֶם לִ֤י סְגֻלָּה֙ מִכָּל־הָ֣עַמִּ֔ים כִּי־לִ֖י כָּל־הָאָֽרֶץ׃ (ו) וְאַתֶּ֧ם תִּהְיוּ־לִ֛י מַמְלֶ֥כֶת כֹּהֲנִ֖ים וְג֣וֹי קָד֑וֹשׁ אֵ֚לֶּה הַדְּבָרִ֔ים אֲשֶׁ֥ר תְּדַבֵּ֖ר אֶל־בְּנֵ֥י יִשְׂרָאֵֽל׃ (ז) וַיָּבֹ֣א מֹשֶׁ֔ה וַיִּקְרָ֖א לְזִקְנֵ֣י הָעָ֑ם וַיָּ֣שֶׂם לִפְנֵיהֶ֗ם אֵ֚ת כָּל־הַדְּבָרִ֣ים הָאֵ֔לֶּה אֲשֶׁ֥ר צִוָּ֖הוּ יְהוָֽה׃ (ח) וַיַּעֲנ֨וּ כָל־הָעָ֤ם יַחְדָּו֙ וַיֹּ֣אמְר֔וּ כֹּ֛ל אֲשֶׁר־דִּבֶּ֥ר יְהוָ֖ה נַעֲשֶׂ֑ה וַיָּ֧שֶׁב מֹשֶׁ֛ה אֶת־דִּבְרֵ֥י הָעָ֖ם אֶל־יְהוָֽה׃ (ט) וַיֹּ֨אמֶר יְהוָ֜ה אֶל־מֹשֶׁ֗ה הִנֵּ֨ה אָנֹכִ֜י בָּ֣א אֵלֶיךָ֮ בְּעַ֣ב הֶֽעָנָן֒ בַּעֲב֞וּר יִשְׁמַ֤ע הָעָם֙ בְּדַבְּרִ֣י עִמָּ֔ךְ וְגַם־בְּךָ֖ יַאֲמִ֣ינוּ לְעוֹלָ֑ם וַיַּגֵּ֥ד מֹשֶׁ֛ה אֶת־דִּבְרֵ֥י הָעָ֖ם אֶל־יְהוָֽה׃ (י) וַיֹּ֨אמֶר יְהוָ֤ה אֶל־מֹשֶׁה֙ לֵ֣ךְ אֶל־הָעָ֔ם וְקִדַּשְׁתָּ֥ם הַיּ֖וֹם וּמָחָ֑ר וְכִבְּס֖וּ שִׂמְלֹתָֽם׃ (יא) וְהָי֥וּ נְכֹנִ֖ים לַיּ֣וֹם הַשְּׁלִישִׁ֑י כִּ֣י ׀ בַּיּ֣וֹם הַשְּׁלִישִׁ֗י יֵרֵ֧ד יְהוָ֛ה לְעֵינֵ֥י כָל־הָעָ֖ם עַל־הַ֥ר סִינָֽי׃ (יב) וְהִגְבַּלְתָּ֤ אֶת־הָעָם֙ סָבִ֣יב לֵאמֹ֔ר הִשָּׁמְר֥וּ לָכֶ֛ם עֲל֥וֹת בָּהָ֖ר וּנְגֹ֣עַ בְּקָצֵ֑הוּ כָּל־הַנֹּגֵ֥עַ בָּהָ֖ר מ֥וֹת יוּמָֽת׃ (יג) לֹא־תִגַּ֨ע בּ֜וֹ יָ֗ד כִּֽי־סָק֤וֹל יִסָּקֵל֙ אוֹ־יָרֹ֣ה יִיָּרֶ֔ה אִם־בְּהֵמָ֥ה אִם־אִ֖ישׁ לֹ֣א יִחְיֶ֑ה בִּמְשֹׁךְ֙ הַיֹּבֵ֔ל הֵ֖מָּה יַעֲל֥וּ בָהָֽר׃ (יד) וַיֵּ֧רֶד מֹשֶׁ֛ה מִן־הָהָ֖ר אֶל־הָעָ֑ם וַיְקַדֵּשׁ֙ אֶת־הָעָ֔ם וַֽיְכַבְּס֖וּ שִׂמְלֹתָֽם׃ (טו) וַיֹּ֙אמֶר֙ אֶל־הָעָ֔ם הֱי֥וּ נְכֹנִ֖ים לִשְׁלֹ֣שֶׁת יָמִ֑ים אַֽל־תִּגְּשׁ֖וּ אֶל־אִשָּֽׁה׃ (טז) וַיְהִי֩ בַיּ֨וֹם הַשְּׁלִישִׁ֜י בִּֽהְיֹ֣ת הַבֹּ֗קֶר וַיְהִי֩ קֹלֹ֨ת וּבְרָקִ֜ים וְעָנָ֤ן כָּבֵד֙ עַל־הָהָ֔ר וְקֹ֥ל שֹׁפָ֖ר חָזָ֣ק מְאֹ֑ד וַיֶּחֱרַ֥ד כָּל־הָעָ֖ם אֲשֶׁ֥ר בַּֽמַּחֲנֶֽה׃ (יז) וַיּוֹצֵ֨א מֹשֶׁ֧ה אֶת־הָעָ֛ם לִקְרַ֥את הָֽאֱלֹהִ֖ים מִן־הַֽמַּחֲנֶ֑ה וַיִּֽתְיַצְּב֖וּ בְּתַחְתִּ֥ית הָהָֽר׃ (יח) וְהַ֤ר סִינַי֙ עָשַׁ֣ן כֻּלּ֔וֹ מִ֠פְּנֵי אֲשֶׁ֨ר יָרַ֥ד עָלָ֛יו יְהוָ֖ה בָּאֵ֑שׁ וַיַּ֤עַל עֲשָׁנוֹ֙ כְּעֶ֣שֶׁן הַכִּבְשָׁ֔ן וַיֶּחֱרַ֥ד כָּל־הָהָ֖ר מְאֹֽד׃ (יט) וַיְהִי֙ ק֣וֹל הַשּׁוֹפָ֔ר הוֹלֵ֖ךְ וְחָזֵ֣ק מְאֹ֑ד מֹשֶׁ֣ה יְדַבֵּ֔ר וְהָאֱלֹהִ֖ים יַעֲנֶ֥נּוּ בְקֽוֹל׃ (כ) וַיֵּ֧רֶד יְהוָ֛ה עַל־הַ֥ר סִינַ֖י אֶל־רֹ֣אשׁ הָהָ֑ר וַיִּקְרָ֨א יְהוָ֧ה לְמֹשֶׁ֛ה אֶל־רֹ֥אשׁ הָהָ֖ר וַיַּ֥עַל מֹשֶֽׁה׃ (כא) וַיֹּ֤אמֶר יְהוָה֙ אֶל־מֹשֶׁ֔ה רֵ֖ד הָעֵ֣ד בָּעָ֑ם פֶּן־יֶהֶרְס֤וּ אֶל־יְהוָה֙ לִרְא֔וֹת וְנָפַ֥ל מִמֶּ֖נּוּ רָֽב׃ (כב) וְגַ֧ם הַכֹּהֲנִ֛ים הַנִּגָּשִׁ֥ים אֶל־יְהוָ֖ה יִתְקַדָּ֑שׁוּ פֶּן־יִפְרֹ֥ץ בָּהֶ֖ם יְהוָֽה׃ (כג) וַיֹּ֤אמֶר מֹשֶׁה֙ אֶל־יְהוָ֔ה לֹא־יוּכַ֣ל הָעָ֔ם לַעֲלֹ֖ת אֶל־הַ֣ר סִינָ֑י כִּֽי־אַתָּ֞ה הַעֵדֹ֤תָה בָּ֙נוּ֙ לֵאמֹ֔ר הַגְבֵּ֥ל אֶת־הָהָ֖ר וְקִדַּשְׁתּֽוֹ׃ (כד) וַיֹּ֨אמֶר אֵלָ֤יו יְהוָה֙ לֶךְ־רֵ֔ד וְעָלִ֥יתָ אַתָּ֖ה וְאַהֲרֹ֣ן עִמָּ֑ךְ וְהַכֹּהֲנִ֣ים וְהָעָ֗ם אַל־יֶֽהֶרְס֛וּ לַעֲלֹ֥ת אֶל־יְהוָ֖ה פֶּן־יִפְרָץ־בָּֽם׃ (כה) וַיֵּ֥רֶד מֹשֶׁ֖ה אֶל־הָעָ֑ם וַיֹּ֖אמֶר אֲלֵהֶֽם׃ (ס)

(1) On the third new moon after the Israelites had gone forth from the land of Egypt, on that very day, they entered the wilderness of Sinai. (2) Having journeyed from Rephidim, they entered the wilderness of Sinai and encamped in the wilderness. Israel encamped there in front of the mountain, (3) and Moses went up to God. The LORD called to him from the mountain, saying, “Thus shall you say to the house of Jacob and declare to the children of Israel: (4) ‘You have seen what I did to the Egyptians, how I bore you on eagles’ wings and brought you to Me. (5) Now then, if you will obey Me faithfully and keep My covenant, you shall be My treasured possession among all the peoples. Indeed, all the earth is Mine, (6) but you shall be to Me a kingdom of priests and a holy nation.’ These are the words that you shall speak to the children of Israel.” (7) Moses came and summoned the elders of the people and put before them all that the LORD had commanded him. (8) All the people answered as one, saying, “All that the LORD has spoken we will do!” And Moses brought back the people’s words to the LORD. (9) And the LORD said to Moses, “I will come to you in a thick cloud, in order that the people may hear when I speak with you and so trust you ever after.” Then Moses reported the people’s words to the LORD, (10) and the LORD said to Moses, “Go to the people and warn them to stay pure today and tomorrow. Let them wash their clothes. (11) Let them be ready for the third day; for on the third day the LORD will come down, in the sight of all the people, on Mount Sinai. (12) You shall set bounds for the people round about, saying, ‘Beware of going up the mountain or touching the border of it. Whoever touches the mountain shall be put to death: (13) no hand shall touch him, but he shall be either stoned or shot; beast or man, he shall not live.’ When the ram’s horn sounds a long blast, they may go up on the mountain.” (14) Moses came down from the mountain to the people and warned the people to stay pure, and they washed their clothes. (15) And he said to the people, “Be ready for the third day: do not go near a woman.” (16) On the third day, as morning dawned, there was thunder, and lightning, and a dense cloud upon the mountain, and a very loud blast of the horn; and all the people who were in the camp trembled. (17) Moses led the people out of the camp toward God, and they took their places at the foot of the mountain. (18) Now Mount Sinai was all in smoke, for the LORD had come down upon it in fire; the smoke rose like the smoke of a kiln, and the whole mountain trembled violently. (19) The blare of the horn grew louder and louder. As Moses spoke, God answered him in thunder. (20) The LORD came down upon Mount Sinai, on the top of the mountain, and the LORD called Moses to the top of the mountain and Moses went up. (21) The LORD said to Moses, “Go down, warn the people not to break through to the LORD to gaze, lest many of them perish. (22) The priests also, who come near the LORD, must stay pure, lest the LORD break out against them.” (23) But Moses said to the LORD, “The people cannot come up to Mount Sinai, for You warned us saying, ‘Set bounds about the mountain and sanctify it.’” (24) So the LORD said to him, “Go down, and come back together with Aaron; but let not the priests or the people break through to come up to the LORD, lest He break out against them.” (25) And Moses went down to the people and spoke to them.

ת"ר בששי בחדש ניתנו עשרת הדברות לישראל רבי יוסי אומר בשבעה בו אמר רבא דכולי עלמא בר"ח אתו למדבר סיני כתיב הכא (שמות יט, א) ביום הזה באו מדבר סיני וכתיב התם (שמות יב, ב) החדש הזה לכם ראש חדשים מה להלן ר"ח אף כאן ר"ח ודכולי עלמא בשבת ניתנה תורה לישראל כתיב הכא (שמות כ, ז) זכור את יום השבת לקדשו וכתיב התם (שמות יג, ג) ויאמר משה אל העם זכור את היום הזה מה להלן בעצומו של יום אף כאן בעצומו של יום כי פליגי בקביעא דירחא רבי יוסי סבר בחד בשבא איקבע ירחא ובחד בשבא לא אמר להו ולא מידי משום חולשא דאורחא בתרי בשבא אמר להו (שמות יט, ו) ואתם תהיו לי ממלכת כהנים
The Sages taught: On the sixth day of the month of Sivan, the Ten Commandments were given to the Jewish people. Rabbi Yosei says: On the seventh day of the month. Rava said: Everyone agrees that the Jews came to the Sinai desert on the New Moon, as it is written here: “In the third month after the children of Israel were gone forth out of the land of Egypt, the same day came they into the wilderness of Sinai” (Exodus 19:1), without elaborating what day it was. And it is written there: “This month shall be to you the beginning of months; it shall be the first month of the year to you” (Exodus 12:2). Just as there, the term “this” is referring to the New Moon, so too, here the term is referring to the New Moon. And similarly, everyone agrees that the Torah was given to the Jewish people on Shabbat, as it is written here in the Ten Commandments: “Remember the Shabbat day to keep it holy” (Exodus 20:8), and it is written there: “And Moses said to the people: Remember this day, in which you came out from Egypt, out of the house of bondage, for by strength of hand the Lord brought you out from this place; there shall be no leaven eaten” (Exodus 13:3). Just as there, the mitzva of remembrance was commanded on the very day of the Exodus, so too, here the mitzva of remembrance was commanded on the very day of Shabbat. Where Rabbi Yosei and the Sages disagree is with regard to the determination of the month, meaning which day of the week was established as the New Moon. Rabbi Yosei held: The New Moon was established on the first day of the week, and on the first day of the week He did not say anything to them due to the weariness caused by the journey. On the second day of the week, He said to them: “And you shall be to Me a kingdom of priests and a holy nation; these are the words that you shall speak to the children of Israel” (Exodus 19:6).
בתלתא אמר להו מצות הגבלה בארבעה עבוד פרישה ורבנן סברי בתרי בשבא איקבע ירחא בתרי בשבא לא אמר להו ולא מידי משום חולשא דאורחא בתלתא אמר להו ואתם תהיו לי בארבעה אמר להו מצות הגבלה בה' עבוד פרישה מיתיבי (שמות יט, י) וקדשתם היום ומחר קשיא לר' יוסי אמר לך ר' יוסי יום אחד הוסיף משה מדעתו דתניא ג' דברים עשה משה מדעתו והסכים הקב"ה עמו הוסיף יום אחד מדעתו ופירש מן האשה ושבר את הלוחות
On the third day of the week, God said to them the mitzva of setting boundaries around Mount Sinai. On the fourth day of the week, the husbands and wives separated from one another. And the Rabbis hold: On the second day of the week the New Moon was established, and on the second day of the week God did not say anything to them due to the weariness caused by their journey. On the third day of the week, God said to them: “And you shall be to Me a kingdom of priests and a holy nation; these are the words that you shall speak to the children of Israel” (Exodus 19:6). On the fourth day of the week, God said to them the mitzva of setting boundaries around Mount Sinai. On the fifth day of the week, the husbands and wives separated from one another. The Gemara raises an objection: Doesn’t the verse state: “And the Lord said to Moses: Go to the people and sanctify them today and tomorrow and let them wash their garments” (Exodus 19:10), indicating that the husbands and wives were separated for only two days? This is difficult according to the opinion of Rabbi Yosei, who said earlier that the separation was for three days. The Gemara answers: Rabbi Yosei could have said to you: Moses added one day to the number of days that God commanded based on his own perception, as it was taught in a baraita: Moses did three things based on his own perception, and the Holy One, Blessed be He, agreed with him. He added one day to the days of separation before the revelation at Sinai based on his own perception. And he totally separated from his wife after the revelation at Sinai. And he broke the tablets following the sin of the Golden Calf.
ת"ר בשעת פטירתו של רבי אמר לבני אני צריך נכנסו בניו אצלו אמר להם הזהרו בכבוד אמכם נר יהא דלוק במקומו שולחן יהא ערוך במקומו מטה תהא מוצעת במקומה יוסף חפני שמעון אפרתי הם שמשוני בחיי והם ישמשוני במותי:
§ The Sages taught: At the time of the passing of Rabbi Yehuda HaNasi, he said: I need my sons. His sons entered his room. He said to them as a last will and testament: Be careful with the honor of your mother. He said further: My lamp should be lit in its usual place, my table should be set in its usual place, and the bed should be arranged in its usual place. Yosef Ḥeifani and Shimon Efrati; they served me during my lifetime and they will serve me in my death.