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Speaker: Matija Bucic (Princeton University and Institute for Advanced Study)23Speaker: Matija Bucic (Princeton University and Institute for Advanced Study)3/10/2022 6:00:00 PM3/10/2022 7:00:00 PM<img alt="" src="/CASEventsImages/CASLockup_Logo_2022.png" style="BORDER:0px solid;" />GP0|#76dcda40-cab3-49f1-997a-050403489abd;L0|#076dcda40-cab3-49f1-997a-050403489abd|Seminar;GTSet|#40f43609-b3a6-498e-b810-f62732269299;GPP|#7ae17a5c-653c-4d0b-96a7-3b708b59814d;GP0|#ad970a66-03c4-4487-8932-f6d1c08a065c;L0|#0ad970a66-03c4-4487-8932-f6d1c08a065c|Natural Sciences;GTSet|#40f43609-b3a6-498e-b810-f62732269299;GPP|#0966b27d-0d92-4427-a0f9-4bff115e8219Seminar<p><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;"><strong>Title:</strong> Tight Ramsey bounds for multiple copies of a graph</span><br style="box-sizing:border-box;color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;"><br style="box-sizing:border-box;color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;"><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;"><strong>Abstrac</strong><strong>t:</strong></span><br style="box-sizing:border-box;color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;"><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">The Ramsey number r(G) of a graph G is the smallest integer n such that any </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">2-colouring of the edges of a clique on n vertices contains a monochromatic </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">copy of G. Determining the Ramsey number of G is a central problem of Ramsey </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">theory with a long history. Despite this there are very few classes of </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">graphs G for which the value of r(G) is known exactly. One such family </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">consists of large vertex disjoint unions of a fixed graph H, we denote such </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">a graph, consisting of n copies of H by nH. This classical result was proved </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">by Burr, Erdős and Spencer in 1975, who showed r(nH)=(2|H|−α(H))n+c, for </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">some c=c(H), provided n is large enough. Since it did not follow from their </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">arguments, Burr, Erdős and Spencer further asked to determine the number of </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">copies we need to take in order to see this long term behaviour and the </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">value of c. More than 30 years ago Burr gave a way of determining c(H), </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">which only applies when the number of copies n is triple exponential in |H|.</span><br style="box-sizing:border-box;color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;"><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">We obtain an essentially tight answer to this very old problem of Burr, </span><span style="color:#2b2b2b;font-family:"segoe ui", "open sans", arial, sans-serif;font-size:14px;background-color:#ffffff;">Erdős and Spencer by showing that the long term behaviour occurs already </span>when the number of copies is single exponential.​​​</p>

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