A WebPage on Van Der Waerden's Theorem

by William Gasarch

(For now this is just papers that I want to gather in one place)

VDW, poly-VDW, HJ, poly-HJ

1. Draft of a book on VDW material by Gasarch and Parrish GPpaper.pdf Purely Combinatorial.

2. Polynomial Extensions of VDW's and Sz's thm, by by Bergelson and Leibman[1]. Has the original proof of Poly VDW thm. BergLeib.pdf, Ergodic Methods.

3. Combinatorial Proofs of the Poly VDW thm and the Poly HJ thm. by Mark Walters[26]. This is the easier proof of Poly VDW. walters.pdf, Purely Combinatorial.

4. A Partition Theorem by Shelah[23]. polyvdwshelah.pdf

5. Set-Polynomials and Polynomial Extensions of the HJ thm. by Bergelson and Leibman[2]. First proof of Poly-HJ. Hard. polyHJ.pdf, Ergodic Theory.

6. Two Combinatorial Theorems on Arithmetic Progressions by Wolfgang Schmidt[22]. This gives some nice lower bounds on VDW numbers. schmidtlowervdw.pdf, Purely combinatorial.

7. Monochromatic Equilateral Right Triangles in the Integer Grid. By Graham and Solymosi[10]. Gets a better upper bounds on W(3,c) as a corollary. graham-solymosi.pdf, Purely combinatorial.

8. A New Method to Construct Lower Bounds for VDW Numbers. By Herwig, Heule, Lamblagen, an Maaren[11]. lower-bds.pdf, Purely Combinatorial.

9. On Sets of Integers Which Contain No Three Terms in Arithmetic Progession. By Salem and Spencer[20]. 3ap-salem.pdf, Purely Combinatorial.

10. On Sets of Integers Not Containing Long Arithmetic Progressiosn. By Laba and Lacey[14]. k-free-sets.pdf, Purely Combinatorial.

11. A Restricted Version of HJ Thm. By Deuber, Promel, Rothchild[7]. restrictedHJ.pdf,

12. An Application of Lovasz Local Lemma-- A New Lower Bound for the van der Waerden Number [25]. by Soltan Szabo. SZABOLOWER.PDF,

13. A construction for partitions which avoid long arithmetic progressions [3] by E. Berlekamp. BERLEKAMPVDW.PDF,

Other Generalizations and Variants of VDW

1. Ramsey's Theorem for $n$-parameter sets. by Graham and Rothchild[9]. A very general from which follows VDW and Ramsey. Graham-Rothchild.pdf, Hard.

2. Note on Combinatorial Analysis. by Richard Rado's This contains both Rado's thm and Gallai-Witt thm. There is both a German version [17] and an English version [18]. rado-gallai-german.pdf, or rado-gallai-english.pdf, Purely Combinatorial.

3. Ein Kombinatorischer Satz der Elementgeometric (German) By Von Ernst Witt[27]. Witt's article that contain Gallai-Witt thm. witt.pdf, Purely Combinatorial but in German.

4. An elementary proof of the canonizing version of Gallai-Witt's theorem by Rödl and Prömel[16]. CanGallaiWittElementary.pdf My notes on this paper: vdwcanNOTES.pdf, Purely Combinatorial.

5. A Canonical Partition Theorem for Equivalence Relations on $Z^n$. Deuber, Graham, Promel, Voigt[6]. VDWcan.pdf, Ergodic theorey or other hard techniques.

6. Restricted Ramsey Configurations. Spencer[24]. res-ram-config.pdf, Purely Combinatorial.

7. VDW's thm on Homothetic Copies of $\{1,1+s,1+s+t\}$. By Kim and Rho [12]. VDWH.pdf,

8. Monochromatic Homothetic Copies of $\{1,1+s,1+s+t\}$[5]. VDWHcopies.pdf,

9. APs in Sequences with Bounded Gaps, by Tom Brown and Donavan Hare[4]. VDWgaps.pdf.

10. The 2-color relative linear VDW numbers by Kim and Rho[13]. VDWlin.pdf.

11. An Infinitary Polynomial VDW Thm. By McCutcheon[15]. infinite-vdw.pdf,

12. Rainbow Arithmetic Progression and Anti-Ramsey Results. By Jungic, Licht, Mahdian, Nesteril, Radoicic. rainbow.pdf,

13. Difference sets without squares. by I.Z. Ruzsa[19]. sqdiff-ruzsa.pdf,

14. On differences of sets of sequences of integers I [21] by Sarkozy. SARKOZYONE.PDF,

Sz's Theorem

1. Tau's exposition of Sz's thm by Tau. tauexpsz.pdf.

2. Notes on Sz's Reg Lemma by Ernie Croot. Good exposition! notesregularity.pdf,

3. A New Proof of Sz's Thm for AP's of Length 4. By Gowers. gowers-sz-4AP.pdf,

4. Roth's Thm on AP's. By Iosevich. notes-roth3ap.pdf,

5. Sz Reg Lemma and its applications in Graph Theory. By Komlos, Simonovitis. szreg-applications.pdf,

6. Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progressions [8] by Hillel Furstenberg. FURSTENBERGSZ.PDF,

7. The Ergodic Theoretic Proof of Sz Thm. By Furstenberg, Katznelson, Ornstein. sz-thm-ergodic-easier.pdf,

8. A New Proof of Sz Thm. By Gowers. sz-thm-gowers-proof.pdf,

9. An alternate proof of Szemeredi's cube lemma using extremal hypergraphs. By Gunderson and Rodl. szcubedensity.pdf,