---

Claim. For the case $k=3$ of Problem 1186, write $V(n)$ for the least number of monochromatic three-term arithmetic progressions over all two-colorings of $\{1,\ldots,n\}$. Theorems 4 and 5 of Pablo A. Parrilo, Aaron Robertson and Dan Saracino, On the asymptotic minimum number of monochromatic 3-term arithmetic progressions, give

$$\frac{1675}{32768}\,n^2(1+o(1))\le V(n)\le\frac{117}{2192}\,n^2(1+o(1)),$$

so $1675/32768\le\delta_3\le117/2192$. The lower bound comes from a Fourier-analytic reduction of the count to a quadratic form, bounded through a semidefinite relaxation with a diagonal shift, closed by checking that an explicit rational matrix is positive definite by computer. The upper bound comes from an explicit coloring in twelve blocks alternating in color. Since $117/2192<1/16$, the upper bound refutes the guess $\delta_3=1/16$, the value of a random coloring. The authors believe the upper bound is sharp. The source card is parrilo_2008_asymptotic_minimum_number_monochromatic_3_term.

Covers. Bounds for $\delta_3$ only: not its exact value, and nothing for $k\ge4$. Carlos Toledo's pending claim that the upper bound is the exact value is on its claim page.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: J. Combin. Theory Ser. A 115 (2008), no. 1, 185--192, doi:10.1016/j.jcta.2007.03.006; the arXiv version was posted 19 September 2006, the date of this page. The site's commentary records both bounds, but the site labels the problem OPEN, so its commentary is not acceptance and no reviewed evidence is listed.