20 research outputs found
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Graphs with no induced K-2,K-t
Consider a graph G on n vertices with αn 2 edges which does not contain an
induced K2,t (t > 2). How large must α be to ensure that G contains, say, a large clique or some fixed subgraph H? We give results for two regimes: for α bounded away from zero and for α = o(1).
Our results for α = o(1) are strongly related to the Induced Tur´an numbers
which were recently introduced by Loh, Tait, Timmons and Zhou. For α bounded away from zero, our results can be seen as a generalisation of a result of Gy´arf´as, Hubenko and Solymosi and more recently Holmsen (whose argument inspired ours)∗Research supported by an EPSRC gran
Graphs with no induced
Consider a graph on vertices with edges which
does not contain an induced (). How large does
have to be to ensure that contains, say, a large clique or some
fixed subgraph ? We give results for two regimes: for bounded away
from zero and for .
Our results for are strongly related to the Induced Tur\'{a}n
numbers which were recently introduced by Loh, Tait, Timmons and Zhou. For
bounded away from zero, our results can be seen as a generalisation of
a result of Gy\'{a}rf\'{a}s, Hubenko and Solymosi and more recently Holmsen
(whose argument inspired ours).Comment: 8 pages; final version incorporating changes suggested by referees;
new result in last sectio
The structure and density of -product-free sets in the free semigroup
The free semigroup over a finite alphabet is the
set of all finite words with letters from equipped with the
operation of concatenation. A subset of is -product-free
if no element of can be obtained by concatenating words from , and
strongly -product-free if no element of is a (non-trivial) concatenation
of at most words from .
We prove that a -product-free subset of has upper Banach
density at most , where . We also determine the structure of the extremal -product-free subsets
for all ; a special case of this proves a conjecture
of Leader, Letzter, Narayanan, and Walters. We further determine the structure
of all strongly -product-free sets with maximum density. Finally, we prove
that -product-free subsets of the free group have upper Banach density at
most , which confirms a conjecture of Ortega, Ru\'{e}, and Serra.Comment: 31 pages, added density results for the free grou
Abundance: Asymmetric Graph Removal Lemmas and Integer Solutions to Linear Equations
We prove that a large family of pairs of graphs satisfy a polynomial
dependence in asymmetric graph removal lemmas. In particular, we give an
unexpected answer to a question of Gishboliner, Shapira, and Wigderson by
showing that for every , there are -abundant graphs of
chromatic number . Using similar methods, we also extend work of Ruzsa by
proving that a set which avoids solutions
with distinct integers to an equation of genus at least two has size
. The best previous bound was and the
exponent of is best possible in such a result. Finally, we investigate
the relationship between polynomial dependencies in asymmetric removal lemmas
and the problem of avoiding integer solutions to equations. The results suggest
a potentially deep correspondence. Many open questions remain.Comment: 28 pages, 4 figure
Reconstructing a point set from a random subset of its pairwise distances
Let be a set of points on the real line. Suppose that each pairwise
distance is known independently with probability . How much of can be
reconstructed up to isometry?
We prove that is a sharp threshold for reconstructing all of
which improves a result of Benjamini and Tzalik. This follows from a
hitting time result for the random process where the pairwise distances are
revealed one-by-one uniformly at random. We also show that is a weak
threshold for reconstructing a linear proportion of .Comment: 13 page
Flashes and Rainbows in Tournaments
Colour the edges of the complete graph with vertex set
with an arbitrary number of colours. What is the smallest integer such
that if then there must exist a monotone monochromatic path of
length or a monotone rainbow path of length ? Lefmann, R\"{o}dl, and
Thomas conjectured in 1992 that and proved this for . We prove the conjecture for
and establish the general upper bound . This reduces the gap between the best lower and upper bounds from
exponential to polynomial in . We also generalise some of these results to
the tournament setting.Comment: 14 page
Product structure of graph classes with bounded treewidth
We show that many graphs with bounded treewidth can be described as subgraphs
of the strong product of a graph with smaller treewidth and a bounded-size
complete graph. To this end, define the "underlying treewidth" of a graph class
to be the minimum non-negative integer such that, for some
function , for every graph there is a graph with
such that is isomorphic to a subgraph of . We introduce disjointed coverings of graphs
and show they determine the underlying treewidth of any graph class. Using this
result, we prove that the class of planar graphs has underlying treewidth 3;
the class of -minor-free graphs has underlying treewidth (for ); and the class of -minor-free graphs has underlying
treewidth . In general, we prove that a monotone class has bounded
underlying treewidth if and only if it excludes some fixed topological minor.
We also study the underlying treewidth of graph classes defined by an excluded
subgraph or excluded induced subgraph. We show that the class of graphs with no
subgraph has bounded underlying treewidth if and only if every component of
is a subdivided star, and that the class of graphs with no induced
subgraph has bounded underlying treewidth if and only if every component of
is a star
31st Annual Meeting and Associated Programs of the Society for Immunotherapy of Cancer (SITC 2016) : part two
Background
The immunological escape of tumors represents one of the main ob- stacles to the treatment of malignancies. The blockade of PD-1 or CTLA-4 receptors represented a milestone in the history of immunotherapy. However, immune checkpoint inhibitors seem to be effective in specific cohorts of patients. It has been proposed that their efficacy relies on the presence of an immunological response. Thus, we hypothesized that disruption of the PD-L1/PD-1 axis would synergize with our oncolytic vaccine platform PeptiCRAd.
Methods
We used murine B16OVA in vivo tumor models and flow cytometry analysis to investigate the immunological background.
Results
First, we found that high-burden B16OVA tumors were refractory to combination immunotherapy. However, with a more aggressive schedule, tumors with a lower burden were more susceptible to the combination of PeptiCRAd and PD-L1 blockade. The therapy signifi- cantly increased the median survival of mice (Fig. 7). Interestingly, the reduced growth of contralaterally injected B16F10 cells sug- gested the presence of a long lasting immunological memory also against non-targeted antigens. Concerning the functional state of tumor infiltrating lymphocytes (TILs), we found that all the immune therapies would enhance the percentage of activated (PD-1pos TIM- 3neg) T lymphocytes and reduce the amount of exhausted (PD-1pos TIM-3pos) cells compared to placebo. As expected, we found that PeptiCRAd monotherapy could increase the number of antigen spe- cific CD8+ T cells compared to other treatments. However, only the combination with PD-L1 blockade could significantly increase the ra- tio between activated and exhausted pentamer positive cells (p= 0.0058), suggesting that by disrupting the PD-1/PD-L1 axis we could decrease the amount of dysfunctional antigen specific T cells. We ob- served that the anatomical location deeply influenced the state of CD4+ and CD8+ T lymphocytes. In fact, TIM-3 expression was in- creased by 2 fold on TILs compared to splenic and lymphoid T cells. In the CD8+ compartment, the expression of PD-1 on the surface seemed to be restricted to the tumor micro-environment, while CD4 + T cells had a high expression of PD-1 also in lymphoid organs. Interestingly, we found that the levels of PD-1 were significantly higher on CD8+ T cells than on CD4+ T cells into the tumor micro- environment (p < 0.0001).
Conclusions
In conclusion, we demonstrated that the efficacy of immune check- point inhibitors might be strongly enhanced by their combination with cancer vaccines. PeptiCRAd was able to increase the number of antigen-specific T cells and PD-L1 blockade prevented their exhaus- tion, resulting in long-lasting immunological memory and increased median survival
The chromatic profile of locally colourable graphs
The classical Andr\'{a}sfai-Erd\H{o}s-S\'{o}s theorem considers the chromatic
number of -free graphs with large minimum degree, and in the case says that any -vertex triangle-free graph with minimum degree greater
than is bipartite. This began the study of the chromatic profile
of triangle-free graphs: for each , what minimum degree guarantees that a
triangle-free graph is -colourable? The profile has been extensively studied
and was finally determined by Brandt and Thomass\'{e}.
Triangle-free graphs are exactly those in which each neighbourhood is
one-colourable. As a natural variant, \L uczak and Thomass\'{e} introduced the
notion of a locally bipartite graph in which each neighbourhood is
2-colourable. Here we study the chromatic profile of the family of graphs in
which every neighbourhood is -colourable (locally -partite graphs) as
well as the family where the common neighbourhood of every -clique is
-colourable. Our results include the chromatic thresholds of these families
as well as showing that every -vertex locally -partite graph with minimum
degree greater than is -colourable.
Understanding these locally colourable graphs is crucial for extending the
Andr\'{a}sfai-Erd\H{o}s-S\'{o}s theorem to non-complete graphs, which we
develop elsewhere.Comment: 26 pages, 18 figures. arXiv admin note: text overlap with
arXiv:2012.1040
Minimum degree stability of -free graphs
Given an -chromatic graph , the fundamental edge stability result
of Erd\H{o}s and Simonovits says that all -vertex -free graphs have at
most edges, and any -free graph with that
many edges can be made -partite by deleting edges.
Here we consider a natural variant of this -- the minimum degree stability of
-free graphs. In particular, what is the least such that any -vertex
-free graph with minimum degree greater than can be made -partite by
deleting edges? We determine this least value for all 3-chromatic
and for very many non-3-colourable (all those in which one is commonly
interested) as well as bounding it for the remainder. This extends the
Andr\'{a}sfai-Erd\H{o}s-S\'{o}s theorem and work of Alon and Sudakov.Comment: 12 pages, 2 figure