Add Sidon set

Author: teorth

README.md

@@ -12,6 +12,7 @@ We are arbitrarily numbering the constants as $C_1$, $C_2$, etc. based on the or
 2. [Erdős minimum overlap constant](https://teorth.github.io/optimizationproblems/constants/2.html)
 3. [A sum-difference constant](https://teorth.github.io/optimizationproblems/constants/3.html)
 4. [Cap set constant](https://teorth.github.io/optimizationproblems/constants/4.html)
+5. [A Sidon set constant](https://teorth.github.io/optimizationproblems/constants/5.html)
 
 ## Maintainers
 

constants/1.md

@@ -14,9 +14,10 @@ for all non-negative $f \colon \mathbb{R} \to \mathbb{R}$.
 | ----- | --------- | -------- |
 | $\pi/2 = 1.57059$ | [SS2002] | |
 | $1.50992$ | [MV2009] | |
-| $1.5053$ | [GGSWT2025] | May 2025 announcement |
-| $1.5032$ | [GGSWT2025] | Dec 2025 preprint release|
-| $1.5029$ | [YKLBMWKCZGS2026] | |
+| $1.5053$ | [GGSWT2025] | May 2025 announcement, AlphaEvolve
+| $1.503164$ | [GGSWT2025] | Dec 2025 preprint release, AlphaEvolve
+| $1.503133$ | [WSZXRYHHMPCHCWDS2025] | ThetaEvolve
+| $1.5029$ | [YKLBMWKCZGS2026] | TTT-Discover
 
 ## Known lower bounds
 
@@ -40,4 +41,5 @@ for all non-negative $f \colon \mathbb{R} \to \mathbb{R}$.
 - [MO2009] Martin, Greg; O’Bryant, Kevin. The supremum of autoconvolutions, with applications to additive number theory. Ill. J. Math. 53, No. 1, 219-235 (2009). [arXiv:0807.5121](https://arxiv.org/abs/0807.5121)
 - [MV2009] Matolcsi, Máté; Vinuesa, Carlos. Improved bounds on the supremum of autoconvolutions. J. Math. Anal. Appl. 372, No. 2, 439-447 (2010). [arXiv:0907.1379](https://arxiv.org/abs/0907.1379)
 - [SS2002] Schinzel, A.; Schmidt, W. M.. Comparison of $L^1$ and $L^\infty$ norms of squares of polynomials. Acta Arith. 104, No. 3, 283-296 (2002).
+- [WSZXRYHHMPCHCWDS2025] Wang, Yiping; Su, Shao-Rong; Zeng, Zhiyuan; Xu, Eva; Ren, Liliang; Yang, Xinyu; Huang, Zeyi; He, Pengcheng; Cheng, Hao; Chen, Weizhu; Wang, Shuohang; Du, Simon Shaolei; Shen, Yelong. ThetaEvolve: Test-time Learning on Open Problems. [arXiv:2511.23473](https://arxiv.org/abs/2511.23473)
 - [YKLBMWKCZGS2026] Yuksekgonul, Mert; Koceja, Daniel; Li, Xinhao; Bianchi, Federico; McCaleb, Jed; Wang, Xiaolong; Kautz, Jan; Choi, Yejin; Zou, James; Guestrin, Carlos; Sun, Yu. [Learning to Discover at Test Time](https://test-time-training.github.io/discover.pdf), 2026.

constants/2.md

@@ -16,8 +16,8 @@ for all non-negative $f,g: [-1,1] \to [0,1]$ with $f+g=1$ on $[-1,1]$ and $\int_
 | $4/9=0.4444\dots$ | Erdős (unpublished) | |
 | $0.4$ | [MRS1956]| |
 | $0.380927$ | [H2016] | |
-| $0.380924$ | [GGSWT2025] | |
-| $0.380876$ | [YKLBMWKCZGS2026] |
+| $0.380924$ | [GGSWT2025] | AlphaEvolve |
+| $0.380876$ | [YKLBMWKCZGS2026] | TTT-Discover |
 
 ## Known lower bounds
 

constants/3.md

@@ -20,7 +20,7 @@ $$ |A-B| \gg |A+B|^{C_3}.$$
 | ----- | --------- | -------- |
 | $1$ | Trivial | |
 | $1.14465$ | [GHR2007] |
-| $1.1479$ | [GGSWT2025] |
+| $1.1479$ | [GGSWT2025] | AlphaEvolve
 | $1.173050$ |[G2025] |
 | $1.173077$ |[Z2025]|
 

constants/4.md

@@ -20,7 +20,7 @@ $C_4$ is the largest constant such that, for large $n$, there exists subsets of
 | $2.2101$ | [CF1994] |
 | $2.2173$ | [E2004] |
 | $2.2180$ | [T2002] |
-| $2.2202$ | [RBNBKDREWFKF2023] |
+| $2.2202$ | [RBNBKDREWFKF2023] | Funsearch
 
 ## Additional comments and links
 
@@ -29,9 +29,9 @@ $C_4$ is the largest constant such that, for large $n$, there exists subsets of
 
 ## References
 
-- [CF1994] Calderbank, A. Robert; Fishburn, Peter C. Maximal three-independent subsets of {0,1,2}n. Des. Codes Cryptogr. 4, No. 3, 203-211 (1994).
-- [E2004] Edel, Y. New lower bounds for caps in AG(4, 3). Des. Codes Cryptogr. 33, No. 1-3, 149-160 (2004).
+- [CF1994] Calderbank, A. Robert; Fishburn, Peter C. Maximal three-independent subsets of $\{0,1,2\}^n$. Des. Codes Cryptogr. 4, No. 3, 203-211 (1994).
+- [E2004] Edel, Y. New lower bounds for caps in $AG(4, 3)$. Des. Codes Cryptogr. 33, No. 1-3, 149-160 (2004).
 - [EG2016] Ellenberg, Jordan S.; Gijswijt, Dion. On large subsets of Fnq with no three-term arithmetic progression. Ann. of Math. (2) 185 (2017), no. 1, 339–343. [arXiv:1605.09223](https://arxiv.org/abs/1605.09223)
-- [P1970] Pellegrino, Giuseppe. Sul massimo ordine delle calotte in S4,3. Matematiche (Catania) 25 (1970), no. 10, 1–9.
+- [P1970] Pellegrino, Giuseppe. Sul massimo ordine delle calotte in $S_{4,3}$. Matematiche (Catania) 25 (1970), no. 10, 1–9.
 - [RBNBKDREWFKF2023] Romera-Paredes, Bernardino; Barekatain, Mohammadamin; Novikov, Alexander; Balog, Matej; Kumar, M. Pawan; Dupont, Emilien; Ruiz, Francisco J. R.; Ellenberg, Jordan S.; Wang, Pengming; Fawzi, Omar; Kohli, Pushmeet; Fawzi, Alhussein. Mathematical discoveries from program search with large language models. Nature. 625 (7995): 468–475 (2023).
 - [T2002] Tyrrell, Fred. New lower bounds for cap sets. Discrete Analysis. 2023 (20). [arXiv:2209.10045](https://arxiv.org/abs/2209.10045).

constants/5.md

@@ -0,0 +1,37 @@
+# A Sidon set constant
+
+## Description of constant
+
+$C_5$ is the smallest constant such that Sidon sets in $\{1,\dots,N\}$ have cardinality $N^{1/2} + (C_5 + o(1))N^{1/4}$.
+
+## Known upper bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $1$ | [ET41], [Li69] | |
+| $0.998$ | [BFR21] | |
+| $0.98183$ | [CHO25] | |
+| $0.97633$ | Carter, Georgiev, Gomez--Serrano, Hunter, O'Bryant, Tao, Wagner ([unpublished](https://terrytao.wordpress.com/2025/11/05/mathematical-exploration-and-discovery-at-scale/#comment-689052), 2025) | AlphaEvolve |
+
+## Known lower bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $0$ | [Si38] | |
+
+## Additional comments and links
+
+- This is part of [Erdős problem #30](https://www.erdosproblems.com/30).
+- A survey of the literature can be found at [OBO4].
+
+
+## References
+
+- [BFR21] Balogh, J. and F\"{u}redi, Z. and Roy, S., An upper bound on the size of Sidon sets. [arXiv:2103.15850](https://arxiv.org/abs/2103.15850) (2021).
+- [CHO25] Carter, D. and Hunter, Z. and O'Bryant, K., On the diameter of finite {S}idon sets. Acta Math. Hungar. (2025), 108--126.
+- [ET41] Erd\H{o}s, P. and Tur\'{a}n, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.
+- [Li69] Lindstr\"{o}m, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.
+- [OBO4] O'Bryant, Kevin, A complete annotated bibliography of work related to {S}idon
+sequences. Electron. J. Combin. (2004), 39.
+- [OBO22] O'Bryant, K., On the size of finite Sidon sets. [arXiv:2207.07800](https://arxiv.org/abs/2207.07800) (2022).
+- [Si38] Singer, James, A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. (1938), 377--385.