Erd\H{o}s-Ko-Rado theorem in Peisert-type graphs

The celebrated Erd\H{o}s-Ko-Rado (EKR) theorem for Paley graphs (of square order) states that all maximum cliques are canonical in the sense that each maximum clique arises from the subfield construction. Recently, Asgarli and Yip extended this result to Peisert graphs and other Cayley graphs which are Peisert-type graphs with nice algebraic properties on the connection set. On the other hand, there are Peisert-type graphs for which the EKR theorem fails to hold. In this paper, we show that the EKR theorem of Paley graphs extends to almost all pseudo-Paley graphs of Peisert-type. Furthermore, we establish the stability results of the same flavor.Comment: 10 pages, typos correcte

Integral point sets over finite fields

We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean spaces $\mathbb{E}^m$. We determine their maximal cardinality $\mathcal{I}(\mathbb{F}_q,2)$. For arbitrary commutative rings $\mathcal{R}$ instead of $\mathbb{F}_q$ or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.Comment: 22 pages, 4 figure

Refined estimates on the clique number of generalized Paley graphs

We show that the clique number of the $d$-Paley graph of order $q$ is at most $\sqrt{q/d}+O(\sqrt{q/p})$, where $q$ is an odd power of a prime $p$. This significantly improves the best-known generic upper bound $\sqrt{q}-o(p)$ and matches with the bound $\sqrt{p/d}+O(1)$ for primes $p$ in a recent breakthrough work of Hanson and Petridis. Moreover, our new bound is asymptotically sharp for an infinite family of graphs, which leads to the further discovery of the first nontrivial instance of families of generalized Paley graphs where the clique number can be explicitly determined. One key ingredient in our proof is a new lower bound on the number of directions determined by a Cartesian product in the affine Galois plane $AG(2,q)$, which is of independent interest.Comment: 11 pages, comments welcom

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Extremal Peisert-type graphs without the strict-EKR property

Let $q$ be a prime power. We study extremal Peisert-type graphs of order $q^2$ without the strict-EKR property, that is, Peisert-type graphs of order $q^2$ without the strict-EKR property and with the minimum number of edges. First, we determine this minimum number of edges for each value of $q$. If $q$ is a square, we show the uniqueness of extremal graph and its isomorphism with certain affine polar graph. Using the isomorphism, we conclude that there is no Hilton-Milner type result for this extremal graph. We also prove the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph. If $q$ is a cube but not a square, we show the uniqueness of extremal graph and determine the number and the structure of non-canonical cliques. Finally, we show such uniqueness result does not extend to all $q$

Restricted sumsets in multiplicative subgroups

We establish the restricted sumset analog of the celebrated conjecture of S\'{a}rk\"{o}zy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the set of nonzero squares in $\mathbb{F}_q$ cannot be written as a restricted sumset $A \hat{+} A$, extending a result of Shkredov. More generally, we study restricted sumsets in multiplicative subgroups over finite fields as well as restricted sumsets in perfect powers (over integers) motivated by a question of Erd\H{o}s and Moser. We also prove an analog of van Lint-MacWilliams' conjecture for restricted sumsets, equivalently, an analog of Erd\H{o}s-Ko-Rado theorem in a family of Cayley sum graphs.Comment: 23 page

Maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

Let $q$ be a square. Van Lint and MacWilliams conjectured that the only subset $A$ of $\mathbb{F}_q$, such that $0,1 \in A$, $|A|=\sqrt{q}$, and $a-b$ is a square in $\mathbb{F}_q$ for all $a,b \in A$, is the subfield $\mathbb{F}_{\sqrt{q}}$. The conjecture was first proved by Blokhuis. In this paper, we discuss the same problem by replacing the set of squares $S$ with a set $D$ which is a union of cyclotomic classes of $\mathbb{F}_q$ satisfying $|D|=|S|$. Different from the case of squares, the subfield $\mathbb{F}_{\sqrt{q}}$ fails to be a valid choice in general. By analyzing the additive and multiplicative structures of cyclotomic classes, we show such a set $A$ has an equal contribution from each cyclotomic class forming $D$. In modern viewpoint, Blokhuis's theorem can be interpreted as the Erd\H{o}s-Ko-Rado (EKR) theorem for Paley graphs of square order. We adopt this viewpoint and study the subspace structure of maximum cliques in the corresponding pseudo-Paley graphs; we also conditionally improve the Delsarte bound on the clique number of several families of pseudo-Paley graphs.Comment: 36 pages, exposition improve

Assessing the influence of solar ultraviolet radiation exposure on the primary immune response to immunisation with a protein antigen in humans

Ultraviolet radiation (UVR) is immunosuppressive, particularly to antigen-specific cell-mediated processes, acting via direct and indirect (e.g. vitamin D-mediated) pathways. This research aimed to examine the influence of acute and cumulative solar UVR exposure, at doses relevant to day-to-day activities, on the primary immune response to immunisation in humans. The Australian Ultraviolet Radiation and Immunity (AusUVI) Study was a prospective, longitudinal, twin-centre immunotoxicological study. Healthy adults were immunised subcutaneously with the T-cell dependent antigen, keyhole limpet haemocyanin (KLH). Acute personal UVR exposure was measured by electronic UVR dosimeter worn on the wrist for ten days centred on the day of immunisation; and by sun diary. Cumulative UVR exposure was quantified by microtopographic analysis of silicone impressions of sun exposed skin. Variables that might confound the association between UVR and vaccine immune response were measured, including serum vitamin D (25(OH)D) level. Participants attended for five study visits over a period of 31 days, with recruitment spread over one year. Immune function outcomes were: anti-KLH IgG1 antibody levels measured by enzyme linked immunoassay; delayed-type hypersensitivity (DTH) response to KLH antigen (reflecting T helper cell-1 (Th1) processes); and quantification of T-helper cell subsets by flow cytometry-based methods. A pilot study trialled many components of the AusUVI Study protocol and immune assays. The AusUVI Study was conducted in the Australian cities of Canberra (35o2'S) and Townsville (19o1'S) from July 2010 to August 2011. Two hundred and twenty two healthy participants were recruited (Canberra: 110; Townsville: 112). Participants' average age was 27.9 years (range: 18 - 40 years) and 63.5% were female. Participants with both parents of northern European ancestry (70.0%) predominated. 25(OH)D levels and personal UVR exposure varied by season and by site of enrolment. Townsville participants had higher 10-day clothing-adjusted UVR exposure compared with Canberra participants (2.5 vs.1.8 standard erythemal dose (SED); p=0.003). Higher cumulative UVR exposure was strongly associated with age, male sex, Townsville residence and northern European ancestry. In multiple linear regression models, anti-KLH IgG1 response at day 21 post-immunisation was associated with age (antibody titre reduced by 1.6% per year of age; p=0.001) and sex (14.6% higher titre in females compared with males (p=0.004)). No association between KLH IgG1 response and acute or cumulative UVR exposure, or serum 25(OH)D levels was demonstrated. Reduced DTH response to KLH recall challenge at day 21 post-immunisation was associated with higher acute UVR exposure on the day prior ('Day 5') to immunisation (p=0.015), and Days 5-8 and 5-9 (p=0.039 and p=0.025, respectively) that spanned the pre- and post-immunisation period. No association with cumulative UVR or serum 25(OH)D levels was demonstrated. Change in T-helper 17 (Th17) cell percentage between pre- and post-vaccination time points differed in direction when comparing the low and high UVR exposure groups (-0.39% vs. 0.31%; p=0.004). In conclusion, acute personal solar UVR exposure, at doses relevant to day-to-day activities, modulated the primary cell-mediated immune responses to KLH immunisation. Cumulative UVR exposure and serum 25(OH)D levels were not associated with immune function outcomes