A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs

The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of H\`{a}n, Person and Schacht who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an $n$-vertex $3$-graph is $\left(\frac{5}{9}+o(1)\right)\binom{n}{2}$. In this paper we improve on this result, giving a family of degree sequence results, all of which imply the result of H\`{a}n, Person and Schacht, and additionally allow one third of the vertices to have degree $\frac{1}{9}\binom{n}{2}$ below this threshold. Furthermore, we show that this result is, in some sense, tight.Comment: 21 page

Selection bias in the M_BH-sigma and M_BH-L correlations and its consequences

It is common to estimate black hole abundances by using a measured correlation between black hole mass and another more easily measured observable such as the velocity dispersion or luminosity of the surrounding bulge. The correlation is used to transform the distribution of the observable into an estimate of the distribution of black hole masses. However, different observables provide different estimates: the Mbh-sigma relation predicts fewer massive black holes than does the Mbh-L relation. This is because the sigma-L relation in black hole samples currently available is inconsistent with that in the SDSS sample, from which the distributions of L or sigma are based: the black hole samples have smaller L for a given sigma or have larger sigma for a given L. This is true whether L is estimated in the optical or in the NIR. If this is a selection rather than physical effect, then the Mbh-sigma and Mbh-L relations currently in the literature are also biased from their true values. We provide a framework for describing the effect of this bias. We then combine it with a model of the bias to make an estimate of the true intrinsic relations. While we do not claim to have understood the source of the bias, our simple model is able to reproduce the observed trends. If we have correctly modeled the selection effect, then our analysis suggests that the bias in the relation is likely to be small, whereas the relation is biased towards predicting more massive black holes for a given luminosity. In addition, it is likely that the Mbh-L relation is entirely a consequence of more fundamental relations between Mbh and sigma, and between sigma and L. The intrinsic relation we find suggests that at fixed luminosity, older galaxies tend to host more massive black holes.Comment: 12 pages, 7 figures. Accepted by ApJ. We have added a figure showing that a similar bias is also seen in the K-band. A new appendix describes the BH samples as well as the fits used in the main tex

On deficiency problems for graphs

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer $t$ such that the join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an $n$-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $r\geq 3$). In this paper we resolve their problem fully. We also give an analogous result which forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of bounded degree and small bandwidth.Comment: 11 page

Ramsey numbers of cycles versus general graphs

The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n,H)=(n-1)(\chi(H)-1)+\sigma(H)$, where $\sigma(H)$ is the minimum possible size of a colour class in a $\chi(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the same should be true already when $n\geq |H|\chi(H)$. We improve this 40-year-old result of Burr by giving quantitative bounds of the form $n\geq C|H|\log^4\chi(H)$, which is optimal up to the logarithmic factor. In particular, this proves a strengthening of the Allen-Brightwell-Skokan conjecture for all graphs $H$ with large chromatic number.Comment: 20 pages, 3 figures. Final version to appear in Forum of Mathematics, Sigm

Tur\'an Colourings in Off-Diagonal Ramsey Multiplicity

The Ramsey multiplicity constant of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labelled copies of $H$ in a colouring of the edges of $K_n$ with two colours. Fox and Wigderson recently identified a large family of graphs whose Ramsey multiplicity constants are attained by sequences of "Tur\'an colourings;" i.e. colourings in which one of the colour classes forms the edge set of a balanced complete multipartite graph. The graphs in their family come from taking a connected non-3-colourable graph with a critical edge and adding many pendant edges. We extend their result to an off-diagonal variant of the Ramsey multiplicity constant which involves minimizing a weighted sum of red copies of one graph and blue copies of another. We also apply the flag algebra method to investigate the minimum number of pendant edges required for Tur\'an colourings to become optimal when the underlying graphs are small cliques.Comment: 48 pages, 2 figure