Erdős matching conjecture for almost perfect matchings

In 1965 Erd?s asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for $s>101k^3$ and $(s+1)k?n<(s+1)(k+\frac{1}{100k})$.     

Spectral strengthening of a theorem on transversal critical graphs

A transversal set of a graph G is a set of vertices incident to all edges of G. The transversal number of G, denoted by $\tau (G)$, is the minimum cardinality of a transversal set of G. A simple graph G with no isolated vertex is called τ-critical if $\tau (G?e)<\tau (G)$ for every edge $e\in E(G)$. For any τ-critical graph G with $\tau (G)=t$, it has been shown that $|V(G)|\le 2t$ by Erd?s and Gallai and that $|E(G)|\le \left(\begin{array}{c}t+1\\ 2\end{array}\right)$ by Erd?s, Hajnal and Moon. Most recently, it was extended by Gyárfás and Lehel to $|V(G)|+|E(G)|\le \left(\begin{array}{c}t+2\\ 2\end{array}\right)$. In this paper, we prove stronger results via spectrum. Let G be a τ-critical graph with $\tau (G)=t$ and $|V(G)|=n$, and let ${\lambda }_1$ denote the largest eigenvalue of the adjacency matrix of G. We show that $n+{\lambda }_1\le 2t+1$ with equality if and only if G is $t{K}_2$, $K_{s+1}\cup (t?s)K_2$, or $C_{2s?1}\cup (t?s)K_2$, where $2\le s\le t$; and in particular, ${\lambda }_1(G)\le t$ with equality if and only if G is $K_{t+1}$. We then apply it to show that for any nonnegative integer r, we have $n(r+\frac{{\lambda }_1}{2})\le \left(\begin{array}{c}t+r+1\\ 2\end{array}\right)$ and characterize all extremal graphs. This implies a pure combinatorial result that $r|V(G)|+|E(G)|\le \left(\begin{array}{c}t+r+1\\ 2\end{array}\right)$, which is stronger than Erd?s-Hajnal-Moon Theorem and Gyárfás-Lehel Theorem. We also have some other generalizations.