BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Andy Zucker (San Diego) DTSTART:20220126T160000Z DTEND:20220126T170000Z DTSTAMP:20260423T020955Z UID:WCS/45 DESCRIPTION:Title: Big Ramsey degrees in binary free amalgamation classes\nby Andy Zucker (S an Diego) as part of Warwick Combinatorics Seminar\n\n\nAbstract\nIn struc tural Ramsey theory\, one considers a "small" structure $A$\, a "medium" s tructure $B$\, a "large" structure $C$ and a number $r$\, then considers t he following combinatorial question: given a coloring of the copies of $A$ inside $C$ in $r$ colors\, can we find a copy of $B$ inside $C$ all of wh ose copies of $A$ receive just one color? For example\, when $C$ is the ra tional linear order and $A$ and $B$ are finite linear orders\, then this f ollows from the finite version of the classical Ramsey theorem. More gener ally\, when $C$ is the Fraisse limit of a free amalgamation class in a fin ite relational language\, then for any finite $A$ and $B$ in the given cla ss\, this can be done by a celebrated theorem of Nesetril and Rodl. Things get much more interesting when both $B$ and $C$ are infinite. For example \, when $B$ and $C$ are the rational linear order and $A$ is the two-eleme nt linear order\, a pathological coloring due to Sierpinski shows that thi s cannot be done. However\, if we weaken our demands and only ask for a co py of $B$ inside $C$ whose copies of $A$ receive "few" colors\, rather tha n just one color\, we can succeed. For the two-element linear order\, we c an get down to two colors. For the three-element order\, $16$ colors. This number of colors is called the big Ramsey degree of a finite structure in a Fraisse class. Recently\, building on groundbreaking work of Dobrinen\, I proved a generalization of the Nesetril-Rodl theorem to binary free ama lgamation classes defined by a finite forbidden set of irreducible structu res (for instance\, the class of triangle-free graphs)\, showing that ever y structure in every such class has a finite big Ramsey degree. My work on ly bounded the big Ramsey degrees\, and left open what the exact values we re. In recent joint work with Balko\, Chodounsky\, Dobrinen\, Hubicka\, Ko necny\, and Vena\, we characterize the exact big Ramsey degree of every st ructure in every binary free amalgamation class defined by a finite forbid den set.\n LOCATION:https://researchseminars.org/talk/WCS/45/ END:VEVENT END:VCALENDAR