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### Recent Posts

- To cheer you up in difficult times 9: Alexey Pokrovskiy proved that Rota’s Basis Conjecture holds asymptotically
- To Cheer you up in Difficult Times 8: Nathan Keller and Ohad Klein Proved Tomaszewski’s Conjecture on Randomly Signed Sums
- Noam Lifshitz: A new hypercontractivity inequality — The proof!
- To cheer you up in difficult times 7: Bloom and Sisask just broke the logarithm barrier for Roth’s theorem!
- To cheer you up in difficult times 6: Play Rani Sharim’s two-player games of life, read Maya Bar-Hillel presentation on catching lies with statistics, and more.
- To cheer you up in difficult times 5: A New Elementary Proof of the Prime Number Theorem by Florian K. Richter
- To cheer you up in difficult times 4: Women In Theory present — I will survive
- To cheer you up in difficult times 3: A guest post by Noam Lifshitz on the new hypercontractivity inequality of Peter Keevash, Noam Lifshitz, Eoin Long and Dor Minzer
- Harsanyi’s Sweater

### Top Posts & Pages

- To cheer you up in difficult times 9: Alexey Pokrovskiy proved that Rota’s Basis Conjecture holds asymptotically
- Test Your Intuition (27) about the Alon-Tarsi Conjecture
- To Cheer you up in Difficult Times 8: Nathan Keller and Ohad Klein Proved Tomaszewski's Conjecture on Randomly Signed Sums
- Updates and plans III.
- To cheer you up in difficult times 7: Bloom and Sisask just broke the logarithm barrier for Roth's theorem!
- Jim Geelen, Bert Gerards, and Geoﬀ Whittle Solved Rota's Conjecture on Matroids
- A sensation in the morning news - Yaroslav Shitov: Counterexamples to Hedetniemi's conjecture.
- Timothy Chow Launched Polymath12 on Rota Basis Conjecture and Other News
- 'Gina Says'

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# Tag Archives: Roth’s theorem

## A Couple Updates on the Advances-in-Combinatorics Updates

In a recent post I mentioned quite a few remarkable recent developments in combinatorics. Let me mention a couple more. Independent sets in regular graphs A challenging conjecture by Noga Alon and Jeff Kahn in graph theory was about the number of … Continue reading

Posted in Combinatorics, Open problems, Updates
Tagged Independent sets in graphs, Roth's theorem
4 Comments

## Roth’s Theorem: Tom Sanders Reaches the Logarithmic Barrier

Click here for the most recent polymath3 research thread. I missed Tom by a few minutes at Mittag-Leffler Institute a year and a half ago Suppose that is a subset of of maximum cardinality not containing an arithmetic progression of length 3. Let . … Continue reading

Posted in Combinatorics, Open problems
Tagged Endre Szemeredi, Jean Bourgain, Klaus Roth, Roger Heath-Brown, Roth's theorem, Tom Sanders
10 Comments

## Around the Cap-Set problem (B)

Part B: Finding special cap sets This is a second part in a 3-part series about variations on the cap set problem that I studied with Roy Meshulam. (The first post is here.) I will use here a different notation than in part … Continue reading

## An Open Discussion and Polls: Around Roth’s Theorem

Suppose that is a subset of of maximum cardinality not containing an arithmetic progression of length 3. Let . How does behave? We do not really know. Will it help talking about it? Can we somehow look beyond the horizon and try to guess what … Continue reading

Posted in Combinatorics, Open discussion, Open problems
Tagged Cap sets, polymath1, Roth's theorem, Szemeredi's theorem
28 Comments

## Pushing Behrend Around

Erdos and Turan asked in 1936: What is the largest subset of {1,2,…,n} without a 3-term arithmetic progression? In 1946 Behrend found an example with Now, sixty years later, Michael Elkin pushed the the factor from the denominator to the enumerator, … Continue reading

Posted in Combinatorics, Updates
Tagged Arithmetic progressions, Roth's theorem, Szemeredi's theorem
11 Comments